Transmitter, receiver and a method for digital multiple sub-band processing

ABSTRACT

Highly efficient digital domain sub-band based receivers and transmitters.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/123,229 filing date Mar. 3, 2014 titled RECEIVER, TRANSMITTER AND AMETHOD FOR DIGITAL MULTIPLE SUB-BAND PROCESSING which claims priorityfrom provisional patent filing date Jun. 10, 2011, Ser. No.61/495,533—both applications being incorporated herein in theirentirety.

BACKGROUND OF THE INVENTION

Modern optical or wireless receivers and transmitters may apply opticalor analog domain sub-band processing in order to enhance throughput.

There is a growing need to provide efficient optical or wirelessreceivers and transmitters of high throughput and low complicity.

SUMMARY OF THE INVENTION

There may be provided a receiver, a transmitter and a method.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter regarded as the invention is particularly pointed outand distinctly claimed in the concluding portion of the specification.The invention, however, both as to organization and method of operation,together with objects, features, and advantages thereof may best beunderstood by reference to the following detailed description when readwith the accompanying drawings in which:

FIG 1A illustrates prior art receivers and transmitters;

FIG. 1B illustrates an example of a receiver; FIG. 1C illustrates anexample of a transmitter

FIGS. 2-4 illustrate spectra of various filters according to variousembodiments of the invention; FIGS. 5A, 5C, 6B, 7A, 7C, 7E, 8A, 8C, 9A,9C, 9D, 10A, 26, 27A, 27B, 28, 29A, 29B, 30, 31, 35, 36 and 41illustrate examples of transmitter modules; FIGS. 5B, 5D, 6A, 7B, 7D,8B, 9B, 9E, 9F, 10B, 10C, 11, 13, 15, 17, 18A, 18B, 18C, 19A, 19B, 19C,20, 21, 22, 23A, 23B, 24A, 24B, 25A, 25B, 32, 33, 34, 37A, 37B, 38, 39,40A, 40B and 42 illustrate examples of receiver modules; FIG. 12illustrates an example of a receiver and a transmitter; FIG. 14illustrates an example of filters; FIG. 16 illustrates a routing fabric;FIGS. 43A-43F illustrate compensation modules; FIGS. 44A and 44Billustrates examples of joint intersymbol interference (ISI) and coarsetiming offset (CTO) modules; and FIG. 45 illustrates a mid-sub bandprocessor.

It will be appreciated that for simplicity and clarity of illustration,elements shown in the figures have not necessarily been drawn to scale.For example, the dimensions of some of the elements may be exaggeratedrelative to other elements for clarity. Further, where consideredappropriate, reference numerals may be repeated among the figures toindicate corresponding or analogous elements.

DETAILED DESCRIPTION OF THE DRAWINGS

In the following detailed description, numerous specific details are setforth in order to provide a thorough understanding of the invention.However, it will be understood by those skilled in the art that thepresent invention may be practiced without these specific details. Inother instances, well-known methods, procedures, and components have notbeen described in detail so as not to obscure the present invention.

The subject matter regarded as the invention is particularly pointed outand distinctly claimed in the concluding portion of the specification.The invention, however, both as to organization and method of operation,together with objects, features, and advantages thereof, may best beunderstood by reference to the following detailed description when readwith the accompanying drawings.

It will be appreciated that for simplicity and clarity of illustration,elements shown in the figures have not necessarily been drawn to scale.For example, the dimensions of some of the elements may be exaggeratedrelative to other elements for clarity. Further, where consideredappropriate, reference numerals may be repeated among the figures toindicate corresponding or analogous elements.

Because the illustrated embodiments of the present invention may for themost part, be implemented using electronic components and circuits knownto those skilled in the art, details will not be explained in anygreater extent than that considered necessary as illustrated above, forthe understanding and appreciation of the underlying concepts of thepresent invention and in order not to obfuscate or distract from theteachings of the present invention.

The following abbreviations and reference numbers are used in thefollowing text and drawings:

ADC=Analog to Digital Converter

DAC Digital to Analog Converter

CD Chromatic Dispersion

FDE Frequency-Domain-Equalizer

EQZ Equalizer

OS OverSampled

FD Frequency Domain

SB Sub-Band

rOS r-fold oversampled

OS Over sampled

LPF Low Pass Filter

IR Impulse Response

CR Carrier Recovery

LMS Least Mean Squared

PDM POL Division Muxing

POL Polarization

MSB Multi Sub-band

DFT-S DFT Spread

Tx Transmitter

Rx Receiver

SC Single Carrier

NS-SC Nyquist Shaped Single Carrier

MSBE Multi Sub-Band Equaliz(ed/ization)

SSC Sub Single-Carrier

IQ In-phase and Quadrature components

IQI IQ Imbalance

CFO Carrier Frequency Offset

CTO Coarse Timing Offset

E&C Estimation and Compensation

FB Filter Bank

PDM Polarization Division Multiplex(ed/ing)

LO Local Oscillator

FTO Fine Timing Offset

SFO Sampling Freq. Offset

CR Carrier Recovery

MSDD Multi.Symbol.Delay.Det

COMP Compensator

DEMOD Demodulator

PN Phase Noise

PROC Processor

COH Coherent

FE Front-end

BE Back-end

CP Cyclic Prefix

SC-FDM Single-Carrier Frequency Division Multiplexing

ISI Inter Symbol Interference

ICI Inter Channel Interference

pnt Points (e.g. in size of (I)FFTs)

 10 Prior art device  11 Upconverter  12 Band pass filter (BPS)  13Adder  14 Downconverter  20 Interpolator  21 BPF, first filter  22Downconverter  23 Circular shift module  24 BPF, second filter  25Decimator  26 Interpolator  27 BPF, first filter  28 Circulat shiftmodule  29 Upconverter  30 Transmitter  31 BPF, second filter  32 Adder 33- Frequency spectrum  45  51 Serial to parallel converter (S/P)  52Fast Fourier transform (FFT) module  53 Inverse fast Fourier transform(IFFT) module  54 Parallel to serial converter and cyclic prefix added(P/S + CP ADD)  55 Serial to paralle converter and cyclic prefix drop(S/P + CP DROP)  56 Parallel to serial converter (P/S)  57 Circularshift and sub-band extractor (CIRC + SB EXTRACT)  58 Zero padder andcircular shift (ZP&CIRC SHIFT)  60 Filter bank, analysis filter bank(FB)  61 Sub-band processor (in transmitter)  62 Filter bank, synthesisfilter bank (FB)  63 Sub-band processor (in receiver)  64 Sub-band OFDMtransmitter  65 Sub-band OFDM receiver  70 Routing fabric  71 Opticalreceiver front end  72 Analot to digital converter (ADC) also includesI-ADC and Q-ADC  73 Routing and combining fabric  74 Single inputmultiple output (SIMO) filter, such as single input dual   output (SIDO)filter or single input quadruple output (SIQO) filter  75 Delay unit  76Outputs  77 Delay unit  78 Splitter  79 Splitting and routing fabric  80Multiple input single output (MISO) filter, such as dual input single  output (DISO) filter or quadruple input single output (QISO) filter  81Polyphase Finite Impluse Response (FIR) filter array  82 QAM demapperand data multiplexor (QAM demapp + data mux)  83 Mis sub-band processor 84 OFDM sub-band processor  85 In phase-quadrate phase imbalance (IQI)compensator  86 Mixer  87 Joint IQI, CFO, CTO, SFO estimator  88 Halfband extractor  89 Multiple input multiple output (MIMO) equalizer  90Polyphase adaptive tracking module  91 MSSD decoder  92 Datademultiplexor and QAM mapper  93 Mid sub-band sparce insert  94 Guardtomes  95 Add into high or low half band  98 Coherent transmitter backend (COH Tx BE)  99 Digital to analog converter (DAC) 101 Joint IQI,CFO, CTO estimator 103 D&C based CTO + CFO joint EST 105 IQI estimator(IQI EST). 106 Multiplier 107 CFO compensator (CFO COMP) 108 CTOcompensator (CTO COMP) 109 Divider 110 t-pointsMinn Algorithm blocks(t-pnt MA) 111 Autocorrelation based metrics 112 Square of absolutevalue module 113 Angle finder 114 Argmax operation module (argmax k) 115IQI + CTO EST 116 Switch 117 Filter, loop filter 118 Local oscillatorlaser (LO laser) 119 Fiber

There is provided a method and system for breaking the digitalprocessing into multiple parallel virtual sub-channels, occupyingdisjoint spectral sub-bands. The optical community is well-used to thisconcept in the optical or analog sub-carrier domains, but it turns outthat it can also be done efficiently in the digital domain. E.g., in ourembodiment, in the optical Receiver (Rx) ASIC DSP, each exemplary 25 GHzWDM channel would be digitally partitioned into M=16 bands of ˜1.6 GHzeach. Remarkably, this digital demultiplexing into sub-bands may beperformed efficiently, with low computational complexity, with neitherspectral guard-bands in-between the 1.6 GHz sub-channels nor withspectral guard-bands between the 25 GHz WDM channels.

We emphasize that the sub-banding scheme is not analog, but it is ratherpurely digital, performed after A/D conversion for each individualchannel in the WDM multiplex, amounting to a second tier of finefrequency division de-multiplexing. Digital sub-banding providesbenefits similar to those obtained in optically generating relativelynarrowband bands (3-6 GHz) within a super-channel structure, an approachrecently increasingly adopted in super-hero OFDM super-channelexperiments [X. Liu and S. Chandrasekhar, SPPCOM'11 (2011).] However ourdigital sub-channel mux requires much simpler, lower cost andenergy-efficient hardware. As our sub-banding realization is digitalrather than analog, we do away with the cumbersome finely spacedmulti-tone generator, and we eliminate a large number of DACs, ADCs,modulators, optical filters, analog optical receivers, which would becustomarily used in transmission of a finely spaced (3-6 GHz)analog-generated super-channel. Nevertheless, we enjoy the full benefitsof having narrowband frequency-flat sub-bands, which are now digitally(de-)muxed. Furthermore, despite efficiently crowding the multiple 1.6GHz sub-bands with zero spectral guard-bands, we are nevertheless ableto maintain a nearly perfect degree of orthogonality between theindividual sub-bands (i.e., eliminate inter-sub-band crosstalk), whichwould have been impossible in a fine-muxed analog/optical generatedsuper-channel.

The means to perform the digital sub-banding in the Tx and especially inthe Rx is the usage of digital signal processing (DSP) structures in thereal-time hardware processors, called Filter Banks. In particular, thisinvention addresses the so-called Uniform Filter Banks, where ‘uniform’here indicates that the sub-bands of the filter-bank are all of the samespectral shape, and are uniformly distributed along the frequency axis,i.e., are regularly spaced. In the sequel we drop the ‘uniform’qualifier, referring to uniform FB whenever we say FB. Moreover, thisinvention refers to oversampled (OS) filter banks. The OS qualifierdescribes FB configurations whereby the sampling rate of each of thesub-bands is higher than the Nyquist rate (difference between thehighest and lowest frequency), whereas the Critically Sampled (CS)qualifier (sometimes referred to as maximally decimated/interpolated)describes the case when the sub-bands sampling rate precisely equals theNyquist rate per sub-band (that is a fraction 1/M of the overall channelbandwidth, where M is the number of sub-bands).

Mathematical notations: Introducing the relevant mathematicalconventions used in this application we shall use the zero-paddingnotation (x[p])^(ZP[p]:L→M) for a finite sequence with support of Lpoints extended by zero-padding to a support of M points. Notice thatthe discrete variable p over which the zero-padding is performed isexplicitly denoted in the ZP[p] superscript, to distinguish fromadditional indexes, if present.

The IDFT and DFT are defined by

X[β]= ^(β)IDFT_(p) ^(M-pnt) {x[p]}=Σ _(p=0) ^(M) x[p]W ^(pβ,)

x[p]= ^(p)DFT_(β) ^(M-pnt) {X[β]}=1/MΣ_(p-0) ^(M) X[β]W ^(−βp)

where the right-subscript and left-superscript explicitly denote theinput and output variables of the transformation, and we introduced thenotation W _(m)≡exp{j2π/M}.

We denote the modulo-D operation by

[x] _(D) ≡x mod D

x=└x/D┘D+[x] _(D),

for which we alternatively adopt either the regular unipolar conventionspecifying the division remainder to be non-negative, 0≦[x]_(D)≦D or wealternatively adopt bi-polar a convention, −D/2<[x]_(D)≦D/2. Notice thatthese definitions also work not only for integer D but also forreal-valued D.

We define the linear shift (delay) operator D_(v) ^(d) {X(v)}≡X(v−d) aswell as the circular shift modulo R operator C_(v) ^([s]) ^(R){X(v)}≡X([v−s]_(R)), defined for a range v₀≦v<v₀+R for some agreed uponv₀, e.g. v₀=0 or v₀=−R/2. We notice that for a periodic function ofperiod R, the linear shift of a periodic function of period P,restricted to a single period of the function starting at v₀ amounts toa circular shift. The shift operators were introduced here in thefrequency domain but the same definitions apply to the continuous timeor discrete-time domains with the frequency variable v replaced by the(discrete) time variable t (k).

Polyphase components of a discrete-time sequence are denoted as follows.The p-th polyphase component modulo P (P integer) of a sequence x[k] isdefined as, x^([p]) ^(p) [k]≡x[kP+p]. Evidently, due to the modulo Poperation in the subscript, there are precisely P polyphase components,which may be conventionally indexed in a unipolar notation p=0,1, . . ., P−1 or alternatively be indexed in bi-polar notation.

A sparse polyphase modulo Q definition is defined as an Q-fold upsampled(zero-filled) and q-delayed version of the conventional p-th polyphase:

x^([q]) ^(↑Q) [k]≡D^(q) {↑Q{x^([q]) ^(Q) [k]}}, p=0, 1, 2, . . . , L−1

where the upsampling is defined as

↑Q{y[k]}={y[0],0,0, . . . , 0, y[1],0,0, . . . , 0, y[3], 0,0, . . . 0,y[4],0 . . . }

and D is the unit delay operator (z⁻¹). Sparse polyphases arenotationally distinguished from the regular polyphases by the presenceof an up-arrow in the superscript [q]_(↑Q), reminding that not only arethe polyphases taken modulo Q, but also Q-1 zeroes are inserted inbetween their samples and the resulting sequence is also delayed by qtime units. The sparse polyphases satisfy the following decompositionfor any sequence:

x[k]=Σ_(q=0) ^(Q 1)x^([q]) ^(↑Q) [k]  (1)

E.g. the sequence X[k]={x[0], x[1], x[2], x[3], x[4], x[5], x[6], x[7] .. . } is the sum of the following sparse polyphases modulo Q=3:

x^([0]) ^(↑3) [k]={x[0], 0, 0, x[3], 0, 0,x[6], 0 . . . }

x^([1]) ^(↑3) [k]={0, x[1], 0, 0, x[4], 0, 0, x[7] . . . }

x^([2]) ^(↑3) [k]={0, 0, x[2], 0, 0, x[5], 0, x[7] . . . }

We also introduce double polyphase components modulo P, Q as follows:

x ^([p]) ^(p) ^([q]) ^(Q) [k]≡(x ^([p]) ^(p) [k])^([q]) ^(Q)[k]=x[P(Qk+q)+p]=x[PQk+Pq+p]

Finally we may combine regular and sparse polyphases:

x ^([p]) ^(p) ^([q]) ^(Q) [k]≡(x ^([p]) ^(p) [k])^([q]) ^(↑Q) [k]=D ^(q){↑Q{x ^([p]) ^(p) ^([q]) ^(Q) [k]}}=D ^(q) {↑Q{x[PQk+Pq+p]}}

Filter bank description: Introducing now the relevant terminology forfilter banks, a synthesis (analysis) FB is essentially a collection of MBPFs sharing a sharing common output (summation node) or common input(splitter) with the said filters (referred to as bank filters) preceded(followed) by L-fold up (down) samplers. When M=L the FB is referred toas critically sampled (CS) whereas when M>L the FB is referred to asoversampled (OS).

In this text we shall label the BPFs by the index β or i, and we shallalternatively use both unipolar index notation 0<β≦M−1 as well asbipolar index notation −M/2+1<β≦M/2, as most convenient in eachcircumstance. The prototype filter refers to the particular filter outthe BPF collection of filters which is positioned at mid-band, in thecenter of the overall channel which contains the M sub-bands. In thecomplex-envelope domain (referred to the center frequency of thechannel) this filter is actually not BPF but is rather LPF. The otherM−1 bank filters in the digital domain are obtained by uniform circularshifts of the prototype filter, corresponding to modulations by harmonictones in the time domain.

In the bi-polar index notation the LPF prototype filter at mid-band isassigned index β=0, whereas the BPFs in the left/right half-band of thechannel are assigned positive/negative β index. For even M, which is ofinterest here, there is the β=0 center index,

$\frac{M}{2} - 1$

negative indexes and

$\frac{M}{2}$

positive indexes out of which

$\frac{M}{2} - 1$

are the mirror images of each other, namely

${\beta = {\pm 1}},{\pm 2},\ldots \mspace{11mu},{\pm ( {\frac{M}{2} - 1} )}$

whereas the last index

$\beta = \frac{M}{2}$

although positive, will be seen to correspond to two half sub-bands onewhich is at the extreme positive frequency and while the other one is atthe extreme negative frequency. In this text we shall label the bankfilters by both unipolar 0<β≦M−1 and bipolar −M/2+1<β≦M/2 index notationas most convenient in each circumstance.

Let g_(β)[k] (h_(β)[k]) denote the impulse response of the β^(th) filterat the Tx (Rx) and let G_(β)(z) (H_(p)(z)) denote the correspondingz-domain transfer functions. In the bi-polar index notation the bankfilter .

at the index β=0 is baseband, i.e. it is an LPF rather than a BPF. Thisfilter coincides with the prototype filter and its impulse response isdenoted g₀[k] (h₀[k]) for the synthesis (analysis) FBs.

As the FBs considered here are always uniform, the frequency responsesof all BPFs have the same spectral shape as that of the prototype LPF,and may be obtained by uniformly spaced spectral shifts of the prototypeLPF, formally described in terms of the circular shift operator definedabove as G_(β)(e^(j2πv/R))=C_(v) ^([βR/M1]) ^(R) G₀(e^(j2πv/R)) where Ris the Nyquist sampling rate and bandwidth of the overall channel.

The frequency shift corresponds in the the time domain to the modulationrelation

${g_{\beta}\lbrack k\rbrack} = {{{g_{0}\lbrack k\rbrack}e^{{j2\pi}\frac{\beta}{M}k}} = {{g_{0}\lbrack k\rbrack}{\overset{\_}{W}}_{m}^{\beta \; k}}}$

A similar relation holds for the analysis bank filters, with G replacedby H. All BPFs are obtained from the LPF prototype by frequency shiftsas follows:

H _(β)(e ^(j2πv/R))=C _(v) ^([βR/M]) ^(R) H ₀(e ^(j2πv/R)).

Thus, the β^(th) BPF TF is obtained from the prototype TF by rightcircular shifting in the frequency domain by, applying a shift equal toβS=βR/M where S=R/M is the spectral extent of each sub-band.

In particular the largest positive shift corresponding to the largestfilter index β_(max)=M/2, is

${\beta_{\max}S} = {{\frac{M}{2} \cdot \frac{R}{M}} = \frac{R}{2}}$

yielding G_(β) _(max) (e^(j2πv/R))=C_(v) ^([R/2]) ^(R) G₀(e^(j2πv/R)).The BPF with the highest index then has center frequency R/2 right atthe high end of the [−R/2, R/2] channel band. Due to the spectralwrap-around of the digital domain transfer function, the spectralsupport of this filter ends up being split between the extreme positiveand negative frequencies: one half of this filter spans the spectralinterval [R/2−S/2, R/2] while the other half spans −R/2, R/2+S/2]. Weshall see that this extreme filter will not be used for carrying data.Rather its split spectral span will be reserved for the spectraltransitions of ADC/DAC analog anti-aliasing/image-rejection filters.

There is provided a system-level applications at the Rx rather than Tx,namely the application OS FBs for high-speed optical transmission, butthe disclosed techniques are also applicable to wireless communication.These novel system level embodiments, especially at the Rx side, buildupon our additional disclosure of inventive embodiments at the subsystemlevel for both the analysis and synthesis OS FBs. Our OS FB embodiments(both analysis and synthesis FBs) have lower real-time computationcomplexity than the structures mentioned above prior art devices for theOS FBs. In particular the OS FBs disclosed in [fred harris] are based onextensive buffer data shifts, having to shuffle data in real-time at theinput/output of the polyphase filter arrays with the data shifts in thebuffers performed at the system clock rate, which is very costly inreal-time ASIC realizations, especially at the high-speed required ofoptical communication. We first describe multiple alternative novelembodiments of the OS FB sub-system structure, disclosing both analysisand synthesis FBs, followed by system-level applications to realizenovel optical receivers and also disclose novel optical Tx-s forsingle-carrier transmission which provide spectrally efficient Nyquistshaping and can be used with the novel optical Rx-s disclosed here, allof which are based on the OS FBs.

FIG. 1A describes a prior art Tx-Rx link formed by a CS synthesis FB atthe Tx, acting as Frequency Division Multiplexer, combining the Msub-bands (sub-channels) in frequency, to form a single channel, whichis M times wider band, to be transmitted over the link to the Rx. The Rxfront-end comprises a CS analysis FB, acting as Frequency DivisionDemultiplexer, separating out the M sub-bands (sub-channels) infrequency to M distinct sub-channel outputs. Notice that as this is aprior art CS system, as the up-sampling ↑M occurring in the synthesis FBas well as the down-sampling ↓M in the analysis FB are by the samefactor M, which is equal to the number of of sub-bands.

In contrast, in FIGS. 1B-1C we present our novel structures of V-fold OSsynthesis and analysis filter bank. In these respective FBs, theup/down-samplers, ↑L and ↓L are by a factor L which satisfies L<M suchthat V≡M/L>1 is the oversampling factor. Here, according to theteachings of this invention, the synthesis FB of FIG. 1C at the Tx ispreceded by specific per-sub-band processing comprising V-foldINTERPOLATORS (post-filtered up-sampling) and frequency up-shifting, andsimilarly the analysis FB of FIG. 1B at the Rx is followed by additionalper-sub-band processing comprising V-fold DECIMATORS (pre-filtereddown-sampling). The interpolation post-filtering occurring inside theinterpolator blocks, and the anti-aliasing pre-filtering occuring in thedecimator blocks are indicated by three wavy lines in our shorthandnotation.

In addition, it is necessary to combine the interpolation/decimationwith digital up-down frequency shift, as shown by the multipliers(modulators) with the linear phase factor W _(M) ^(±βLk) where k isdiscrete-time, β is the index labeling the sub-band and thecorresponding BPF (branch of the filter bank).

Back to the CS FB structure of FIG. 1A, it is readily seen that sincethe synthesis-analysis FBs act as frequency domain mux-demux, then forback-to-back operation of the Tx-Rx cascade, the simplest near-perfectreconstruction condition is that the BPF-s of the CS FB be shaped asfrequency contiguous brick-wall filters, each of bandwidth B/M, where Bis the total channel bandwidth. This perfect sharpness filteringcondition on the frequency responses of the BPF is very stringent and inpractice when using CS filter-banks, spectral guardbands may areinserted between the sub-bands, incurring tradeoffs between spectralefficiency, inter-sub-channel interference and the computationalcomplexity of the filters, since sharper filters (closer to brickwall)require many more taps to realize.

Now, for the OS FB structure of FIGS. 1B-1C is to provide ideallyperfect reconstruction, according to the teachings of this invention itturns out that it is no longer necessary that the FBs be perfectbrickwall. It now suffices that the frequency responses of the bankfilters each have their respective passbands flat over the extent of thecorresponding sub-bands, however it is no longer necessary that thespectral transitions (from passband to stopband) of the filters bebrickwall. Each of the BPF frequency responses still ought to be flatover its sub-band, i.e. the pass-band must coincide with the sub-band,however the transitions may occur in the neighboring sub-bands—thefilter frequency responses are now allowed to more mildly slope down,yielding significant reduction of the number of taps required to realizethe BPF filters of the FBs, relative to the nearly brick-wall filtersrequired for the CS FBs of FIG. 1A. However, to compensate for the lesssharp BPFs, operating at the high input rate of the overall channels, wemust provide sharp filtering elsewhere in the system, actually in a moreconvenient location, namely in the sub-band processing section. Thepost-/pre-filters of the interpolators/decimators must now be brickwall(which will be seen to be an easier task), such that the spectral imagesarising in up-sampling and down-sampling be nulled out near-perfectly.While it seems that we have just shifted the sharp filtering challengefrom one domain (the overall channel level) to another one (the sub-bandprocessing level), nevertheless the sharp spectral shaping turns outeasier to realize at the sub-band level, and its added computationalcomplexity (relative to the CS structure where no such processingexists) is actually smaller than the computational savings attained forthe BPFs due to the mild sloping characteristics of the BPFs inside theVxOS FBs themselves.

FIGS. 2-4 describe the principle of operation of the novel scheme ofFIGS. 1B-1C in the frequency domain, exemplified for a particular valueof V=M/L of 2,4 and 4/3, which are also of interest in practicalhardware applications. To understand the spectral manipulations, recallthat the upsampling operation ↑L is described in the FD as an M-foldreplication of the basic spectral period of the input digital signal,whereas the downsampling operation ↓L is described (up to a constant) bytaking the input spectrum of spectral support R (equal to the samplingrate of the input signal) and uniformly shifting it at intervals R/L inorder to generate L spectral images, and superposing these images.Following the signal flow for the β^(th) sub-band in FIG. 1C from leftto right, this sub-band input signal has spectral support S≡R/M, where Ris the total channel bandwidth and M is the number of sub-bands. Afterthe ↑V op, the sub-band spectrum is replicated V times. The sharp imagerejection LPF (indicated by the three wavy lines in the interpolatorblock, blocks V-1 of these images, retaining just original basebandspectrum of the sub-band signal s_(β)[k], but resulting in a nullspectral guardband around the baseband spectrum. The overall spectralperiod of this V-fold interpolated ‘digital’ spectrum is now VS. Nextthe freq. up-shifter (modulation by W_(M) ^(βLk)=W_(V) ^(βk))circularlyright shifts this periodic spectrum by a fraction β/V of its spectralperiod, i.e. by

${{VS} \cdot \frac{\beta}{V}} = {\beta \; {S.}}$

Measured in terms of the sub-band bandwidth, S, the shift is an integernumber of units, β, however measured in terms of the over-sampled band,VS, the shift is by a rational number β/V.

The frequency shift operator corresponding to freq. up-conversion, isformally described as C_(v) ^([βS]) ^(VS) =C_(v) ^([[β]) ^(v) ^(s])^(VS) .

In words, we evaluate [β]_(v), and this yields the number of sub-bandspectral units by which the baseband spectrum is shifted to the right atthe output of the first block in the Tx, i.e. at the β^(th) input portof the V-fold OS synthesis FB.

We shall exemplify the spectral analysis for three cases of interest,namely V=2, V=4 and V=4/3, the spectral analyses for which are shown inFIGS. 2-4.

For V=2 the spectral shift at the V-fold OS synthesis input is[[β]₂S]_(2S), i.e. for even β([β]₂=0) we have [[β]₂S]_(2S)=[0·S]_(2S)=0i.e. there is no shift, whereas for odd β we have [[β]₂S]_(2S)=[S]_(2S),i.e. the shift is by one SB unit, i.e. half the 2S oversampled band,turning the spectrum from low-pass to high-pass, as shown in FIG. 2.

For V=4 (FIG. 3) the spectral shift at the V-fold OS synthesis input is[[β]₄S]_(4S), i.e., for the four polyphases of β modulo 4, referred toas quad-0, quad-1, quad-2, quad-3, we have the respective circularshifts [0]_(4S), [S]_(4S), [2S]_(4S), [3S]_(4S) of the 4S oversampledband.

For V=4/3 (FIG. 4) the spectral shift at the V-fold OS synthesis inputis [[β]_(4/3)S]_(4/3S). As a cycles through to 0,1,2, . . . , 15[β]_(4/3) yields

{0, −(1/3), −(2/3), 1/3, 0, −(1/3), −(2/3), 1/3, 0, −(1/3), −(2/3), 1/3,0, −(1/3), −(2/3), 1/3}

Thus, for β=0, 1, 2, 3 the shifts are respectively [0]_(4/3S),[−1/3S]_(4/3S), [−2/3S]_(4/3S), [1/3S]_(4/3S), and then the cyclerepeats.

Analytically considering a general V-value, still following the β^(th)branch, this sparse oversampled band of extent VS enters the ↑Lupconverter and is spectrally replicated L times. The L identicalspectral images form an extended period of LVS=MS=R, occupying the fullchannel bandwidth.

At this point there comes the BPF, carving out one spectral image out ofthe L images. The β^(th) BPF has its passband right over the β^(th) SB,with center at position βS from the spectral origin.

The i-th oversampled band image has its center frequency at positioniVS. The index i=i _(β) of the spectral image capturing the center ofthe β^(th) sub-band is determined from the condition that−1/2VS≦βS−iVS<1/2VS. Writing βS=i_(β)·VS+[βS]_(VS) where

${i_{\beta} \equiv \lfloor \frac{\beta \; S}{VS} \rfloor} = \lfloor \frac{\beta}{V} \rfloor$

or equivalently [βS]_(VS)=βS−i_(β)·VS and recalling from the bi-polarmodulo definition that we must have −1/2VS≦[βS]_(VS)<1/2VS, it followsthat we do satisfy −1/2VS≦βS−i_(β)·VS<1/2VS, i.e. the particular index

${i_{\beta} \equiv \lfloor \frac{\beta \; S}{VS} \rfloor} = \lfloor \frac{\beta}{V} \rfloor$

is the one sought for the spectral image which captures the center ofthe β^(th) sub-band. Moreover, we have [βS]_(VS)=βS−i_(β)·VS, meaningthat the shift between the center of the β^(th) sub-band and the centerof the i_(β) image capturing the sub-band center is precisely[βS]_(VS)=[[β]_(V) S]_(VS). However, this is precisely the spectralshift which was applied at baseband by the up-converting modulator.Therefore, as the ↑L op rigidly shifted the basedband oversampled bandto the i_(β) ^(th) spectral image, then the spectral shift which wasapplied at baseband precisely brings the baseband sub-band right intothe passband of the β^(th) BPF. Thus each of the up-shifted sub-bands iscarved with nearly no distortion (ideally without distortion if thefilter passband is perfectly flat), and all the shifted sub-bands aresuperposed by the adder terminating the V-fold OS synthesis FB. Amultiplexed spectrum of M bands, occupying a spectral extent ±MS/2=±R/2has been generated by the Tx.

Conversely, in the Rx (FIG. 1B) we have the V-fold OS analysis filterbank applying reverse transformations as follows:

The multiplexed spectrum, juxtaposing the M sub-bands in frequency, ispresented in parallel to the filter bank of M BPFs, each of which carvesa corresponding sub-band, with high precision over the extent of thesub-band (as each BPF is flat over a sub-band interval) however as thefilters are allowed to mildly slope down in the two neighboringsub-bands around the pass-band sub-band, then each BPF also picks upout-of-sub-band spectral interference, which nevertheless is wellseparated from the targeted sub-band. The spectral support of each BPFis designed to be precisely ±VS/2 around its center frequency, such thatthe (one-sided) pass-band frequency is S/2, the stop-band is VS/2, andthe transition band occurs over the interval VS/2−S/2=(V−1)S/2. Thespectral rolloff factor, α≡stopband/passband=V, i.e., it equals theoversampling factor. The larger the oversampling factor the milder thefilter and less coefficients it requires. However larger oversamplingfactor means the filter must operate faster, therefore the computationalcomplexity per unit time of each BPF turns out to be invariant in the OSfactor, V.

The spectral support delineated by +/− the stop-band frequency of theBPF around the BPF center frequency then occupies an oversampled band ofextent VS. This band is manipulated by the ↓L down-sampling, in essencereplicated L times and the images are superposed. These images areshifted at regular intervals equal to B/L=MS/L=VS, which are seen toequal to the bandlimitation VS of the identical images to (traced to theoriginal spectrum having been bandlimited by the BPF to VS two-sided).These spectral images then fit together with no aliasing overlapcovering the full channel spectral region of L·VS=MS=R. However,consistent with the VS-periodicity of the superposition of shiftedimages, the sampling rate at the down-sampler output now becomes VS,indicating that we have oversampling by a factor V relative to theoutput of the M-fold downsampling in the CS FB structure of FIG. 1A.

The VS-wide spectrum, which was replicated all the way down to aroundthe origin, includes within it (possibly offset from the origindepending on the BPF index, β and the OS factor, V) the clean sub-bandwith spectral interference around it. It is the role of the down-shifteddecimator final stage to extract the clean sub-band and reject theinterference. To this end, the spectrum is first shifted to baseband bythe frequency down-shifter (multiplication by W_(M) ^(−βLk)=W_(V)^(−βk), then the sharp anti-aliasing filter of the decimator removes thespectral interference around the desired sub-band which now falls atbaseband. Finally the decimator replicates and superposes this spectrumV times, filling up the entire oversampled band of extent VS by V imagesof the sub-band, consistent with the sampling rate being reduced from VSdown to S, which is precisely the Nyquist rate (equal to the two-sidedbandwidth S or twice the one-sided bandwidth S/2). We notice that theoperations occurring in the analysis FB in the Rx are the preciseopposites obtained by time reversal, relative to the operations whichoccurred in the synthesis FB in the Tx. Thus the spectral manipulationsof FIGS. 2-4 shoud be read from the top down for the OS synthesis FBoperation and from the bottom up for the OS analysis FB operation.

FIGS. 5A-5D address the efficient realization of the V-fold interpolatorand decimator, also incorporating the up/down conversion operations. Forrelatively low values of V, a highly efficient high precisionrealization of these operations is based on cascades of IFFT and FFTback to back, achieving near-ideal interpolation and decimation withsharp rejection of images, provided some cyclic prefix extension isallowed as compatible with communication applications. The up/downfrequency shifting functionality (multiplications by W _(M) ^(=βLk)=W_(V) ^(±βk)) shown in FIGS. 1B-1C is implemented here not as time-domainmodulation but equivalently as circular shifting in the frequencydomain, at the interpolator IFFT input in the Tx and the decimator FFToutput in the Rx.

In detail, ignoring the frequency-shifting initially, in FIG. 5A aninterpolator is realized as an N-pnt FFT feeding a VN-pnt IFFT as shownwith N/2 upper outputs of the N-pnt FFT being routed to the uppermostN/2 points of the VN-pnt IFFT, which points represent positivefrequencies, while the N/2 lower outputs of the N-pnt FFT being routedto the lowermost N/2 points of the VN-pnt IFFT, which points representnegative frequencies. In between the two inserted N/2-pnt bands thereare VN-V inputs into the VN-pnt IFFT which are zero-padded.

In the realization of the V-fold decimator of FIG. 5B the oppositeprocess occurs, with the VN-V midpoints of the VN-pnt FFT beingdiscarded (corresponding to anti-aliasing low-pass filterings, as thesepoints represent the high-pass frequencies) and then two N/2 bands atthe beginning and end of the VN-pnt FFT output are routed to the upperand lower halves of the N-pnt IFFT input as shown.

In FIGS. 5C,D we enhance the treatment to combine theinterpolation/decimation with frequency up/down-shifting. In between theIFFT and FFT we place simple routing fabric modules, performing thefollowing DSP functions:

For the modified interpolator at the Tx side (FIG. 5C), the routingfabric inserted in between the N-pnt FFT and the VN-pnt IFFT, calledZERO-PAD & CIRC SHIFT, performs zero-padding of the N-pnt FFT output toVN points, just as described in FIG. 5A, but now followed by a circularshift by [β]_(V)N points of the VN-pnt resulting vector. The circularlyshifted vector is applied to the VN-pnt IFFT input.

For the modified decimator at the Rx side (FIG. 5D), the routing fabricinserted in between the VN-pnt FFT and the N-pnt IFFT, called CIRC&SHIFT& SB EXTRACT, performs a circular shift by [−β]_(V)N points of theVN-pnt output vector of the IFFT, followed by the inverse ofzero-padding, namely extracting out the sub-band by stripping part ofVN-pnt of the vector yielding an N-pnt vector which is applied to theN-pnt IFFT.

Actually, these routing fabrics, and their associated circular shifts donot perform actual shuffling of data, which would be cumbersome here wehave no buffering and no data transfer from location to location withina buffer when we refer to ‘circular shift’, but there is just mapping ofinputs to outputs, achieved by a static permutation of wires connectingthe inputs or outputs of the smaller N-pnt (I)FFT to a particularsub-segment of the VN-pnt input or output of the larger VN-pnt (I)FFT.

Notice that if these two routing fabrics (the ZERO-PAD & CIRC SHIFT andthe CIRC&SHIFT & SB EXTRACT) were cascaded back-to-back, then they wouldresult in an identity system, as the Rx routing fabric is atime-reversed version of the Tx-routing fabric, i.e., the Rx routingfabric may be obtained by time-reversed traversing the Tx-routing fabricin the opposite sense. Next we describe the particular Tx (synthesis FB)routing fabrics for the special cases V=2,4/3,4. The correspondingrouting fabrics for the Rx (analysis FB) are simply time-reversed ofversions of the Tx ones.

We mention that in all these embodiments additional half-band shiftsexchanging the positive and negative frequencies of the (I)FFTs inputsor outputs could be applied at the Tx, provided their mirror images willalso be applied at the Rx in the corresponding (I)FFTs. These half-bandshifts will result in different mappings of the routing fiber which arenevertheless equivalent, as the pairs of half-band shifts at the Tx andRx will cancel each other in the overall cascade. In the specificexamples to follow we do not incorporate these additional half-bandshifts, nevertheless corresponding versions with different routingfibers may also be readily derived by applying such additional half-bandshifts.

For the special case of V=2, the circular shifts at the Tx (FIG. 5C) areby [β]_(V)N=[β]₂N points, i.e. by 0 points (identity) for even β and byN points for odd β. The Tx routing fabric then consists of inserting theoutput of the N-pnt FFT into either the low-pass half (for even β branchof the FB) or high-pass half (for odd β branch of the FB) of theVN=2N-pnt IFFT. In more detail, let us partition the 2N-pnt input of theIFFT into four quarters each containing N/2-pnt. For even β there is nocircular shift, thus mapping is precisely as described in FIG. 5A(taking the VN-pnt IFFT as 2N-pnt IFFT), i.e. into the top and bottomquarters of N/2 points of the 2N-pnt IFFT input, which may becharacterized as the low-pass band.

For odd β, as the circular shift becomes N, the two N-pnt FFT outputhalves are now mapped into the two middle segments of the 2N-pnt IFFTinput (which were previously zero-padded in the case of even β), whichmay be characterized as the high-pass band. Specifically labeling thefour quarters of the of the 2N-pnt IFFT input from the top down.by theindexes 0,1,2,3, then in the Tx, for even β, the top N/2-points of theN-pnt FFT output are mapped into quarter-0, whereas the bottomN/2-points of the N-pnt FFT output are mapped into quarter-3. At the Tx,for odd β the top N/2-points of the N-pnt FFT output are mapped intoquarter-2, whereas the bottom N/2-points of the N-pnt FFT output aremapped into quarter-1.

Dually, at the Rx (the sub-band processors for the analysis FB), theflow is reversed: for even β, quarter-0 of the 2N-pnt FFT output ismapped into the top N/2-points of the N-pnt IFFT output, quarter-3 ofthe 2N-pnt FFT output is mapped into the bottom N/2-points of the N-pntIFFT. For odd β quarter-2 of the 2N-pnt FFT output is mapped into thetop N/2-points of the N-pnt IFFT output, quarter-1 of the 2N-pnt FFToutput is mapped into the bottom N/2-points of the N-pnt IFFT.

Overall, for V=2 (twice OS FB), the Tx routing fabric may becharacterized as ADD into HIGH/LOW HALF-BAND, whereas the dual Rxrouting fabric may be characterized as EXTRACT HIGH/LOW HALF-BAND.

For V=4/3 at the Tx (synthesis FB), the circular shifts are by[β]_(V)N=[β]_(4/3)N points. Since {[4β′+0]_(4/3), [4β′+1]_(4/3),[4β′+2]_(4/3), [4β′+3]_(4/3)}={0,1,2/3,1/3} then the right shiftsapplied by the circular fabric are by the following number of points:

-   For β mod 4=0 by 0 points (no circular shifts); For β mod 4=1 by    N-pnt;-   For β mod 4=2 by 2/3 N-pnt; For β mod 4=3 by 1/3 N-pnt;

Further incorporating the zero-padding operation, the resulting cascadeof zero-mapping and the circular shifts above may be described asfollows: Partition the N-pnt FFT output into 6 sixths (each of N/6-pnt)and partition the 4/3 N-pnt IFFT input into 8 eights (each also ofN/6-pnt). Then, labeling the 6 sixths from top to bottom as 0,1, . . . ,5 and labeling the 8 eights from top to bottom as 0,1, . . . , 7, the 6sixths are mapped into the 8 eights, including zero-padding of two ofthe 8 eights, as described next.

-   For β mod 4=0, the sixths 0,1,2 are mapped into eights 0,1,2 while    sixths 3,4,5 are mapped into eights 5,6,7.-   For β mod 4=1 we apply a circular shift of N-pnt (i.e. 6 mod 8    eights) to the eights obtained for β mod 4=0. Thus, sixths 0,1,2 are    mapped into eights 6,7,8 while sixths 3,4,5 are mapped into eights    3,4,5.

For β mod 4=2 we apply a circular shift of 2/3 N-pnt (i.e. 4 mod 8eights) to the eights obtained for β mod 4=0. Thus, sixths 0,1,2 aremapped into eights 4,5,6 while sixths 3,4,5 are mapped into eights1,2,3.

For β mod 4=3 we apply a circular shift of 1/3 N-pnt (i.e. 2 mod 8eights) to the eights obtained for β mod 4=0. Thus, sixths 0,1,2 aremapped into eights 2,3,4 while sixths 3,4,5 are mapped into eights7,0,1. This concludes the complete description of the routing fabric atthe Tx for V=4/3. As for the Rx side, the mappings are simply reversedwith the 8 eights of the 4N/3-FFT output being mapped into the 6 sixthsof the N-pnt IFFT input (after dropping two of the eights).

For V=4 at the Tx, circular shifts are by [β]_(V)N=[β]₄N points. Thenthe right shifts applied by the circular fabric are by the followingnumber of points: For β mod 4=0 by 0 points (no circular shifts); For βmod 4=1 by N-pnt;

-   For β mod 4=2 by 2N-pnt; For β mod 4=3 by 3N-pnt;

Further incorporating the zero-padding operation, the resulting cascadeof zero-mapping and the circular shifts above may be described asfollows: Partition the N-pnt FFT output into two N/2-pnt halves andpartition the 4N-pnt IFFT input into 8 eights (each also of N/2-pnt).Then, labeling the two halves from top to bottom as 0,1 and labeling the8 eights from top to bottom as 0,1, . . . , 7, the 2 halves are mappedinto the 8 eights, including zero-padding of 6 of the 8 eights, asdescribed next.

For β mod 4=0 the halves 0,1 are mapped into eights 0,7. For successivevalues of β mod 4 in increments of unity the output eights 0,7 areshifted by N over the 4N record, i.e. by 2 mod 8 eights at a time.

For β mod 4=1 the halves 0,1 are mapped into eights 2,1. For β mod 4=2the halves 0,1 are mapped into eights 4,3 . . . . For β mod 4=3 thehalves 0,1 are mapped into eights 6,5.

This concludes the complete description of the routing fabric at the Txfor V=4. As for the Rx side, the mappings are simply reversed with the 8eights of the 4N-FFT output being mapped into the two halves of theN-pnt IFFT input being mapped into the two halves (after dropping 6 ofthe eights).

Having described the frequency shifted interpolator decimatorrealizations in FIG. 5 we substitute these realizations in theend-to-end diagram of FIG. 1B resulting in the overall realization ofthe FB synthesis-analysis cascade of FIG. 6.

FIGS. 6A-6B describe the preferred realization of end-to-end thesynthesis-analysis FB based link, which may be used to transmit inparallel over a total bandwidth B a collection of M tributarysub-channels, each of bandwidth B/M, carrying any type of modulation.FIG. 6A describes the Rx side which comprises an analysis filter bankfollowed by sub-band processors, while FIG. 6B describes the Tx side,which comprises a synthesis filter bank preceded by sub-band processors.The sub-band processors realize the frequency down-/up-shifteddecimation/interpolation by means of back to back (I)FFTs with routingfabric.

FIG. 7A-7E address the special case of transmission of tributarysub-channels bearing the OFDM modulation format, through a Txaggregating these OFDM tributary sub-channels by means of an OSsynthesis FB, and a Rx demultiplexing thee individual OFDM tributarysub-channels by means of an OS analysis FB. In FIG. 7A sub-band OFDMTx-s are placed on each branch of the OS synthesis FB of FIG. 6B andlikewise sub-band OFDM Rx-s are placed on each branch of the OS analysisFB of FIG. 6A. The inserted sub-band OFDM Tx-s are stylized here to justshow the essential N-pnt IFFT functionality and likewise the sub-bandOFDM Rx-s are represented by the essential N-pnt FFT functionality. Thesize N of the OFDM (I)FFTs is selected to equal that of the outer (I)FFTfeaturing in the V-fold interpolation and decimation modules. As aresult, the OFDM (I)FFT at each end gets cancelled by the corresponding(I)FFT in the interpolation or decimation block, as shown. The resultingsimplified diagrams for the Tx and Rx are shown in FIG. 7C,D. Now thereis a single VN-pnt (I)FFT remaining in the up-/down-frequency-shiftedinterpolation/decimation module, which is now conceptually detached fromthe V-fold synthesis/analysis FB, rather viewed as part of the sub-bandOFDM Tx/Rx. Finally, in FIG. 7E we detail the main functions of thesub-band OFDM Tx and Rx (top and bottom parts of the figurerespectively). Additional conventional functions in the OFDM Tx and Rxwhich are not relevant to the FB sub-banding concept are not shown.

The sub-band OFDM Tx (top of FIG. 7E) features a 1:N S/P conversionfollowed by the mapping of inputs similar to that shown in FIG. 5D, thenfollowed by a right circular shift, then a VN-pnt IFFT, then theaddition of cyclic prefix (CP) and P/S to generate the output. The OFDMRx essentially performs these functions in reverse, as shown in thebottom part of FIG. 7E.

FIGS. 8A-8C,9A-9F,10A-10D show embodiments applying the OS FBs for

OFDM transmission, in effect detailing the general OFDM Tx/Rx structureof FIG. 7B,C for specific OS parameter values of typical interest: V=2,4/3, 4.

FIGS. 8A-8C address the V=2 OS FB based OFDM Tx/Rx, comprising the twiceoversampled FBs and arrays of sub-band OFDM Tx-s and Rx-s as shown inFIGS. 8A and 8 b.

FIG. 8C details the structure of the routing fabric in the sub-band OFDMTx (top two parts of the figure) and sub-band OFDM Rx (bottom two partsof the figure), for even and odd values of the sub-band (or FB port)index β. Notice that relative to our earlier description of routingfabric for V=2, here we apply extra half-band shifts interchanging thetwo mapped halves, which is more convenient yet equivalent to ouroriginal specification of the routing fabric for V=2.

Thus, the routing fabric of the sub-band Tx/Rx for V=2 amounts to addingthe vector of N OFDM sub-carriers into the high and low half bands ofthe twice oversampled band of 2N points which is mapped onto the IFFTinputs at the Tx (with an opposite process of extracting half of the 2Nband (therefore dropping the complementary half) in order to down-sampleby a factor of 2, transitioning from 2N points to N points.

FIGS. 9A-9F address the V=4/3 OS FB based OFDM Tx/Rx. FIG. 9A-9B showsthe generic sub-band OFDM Tx/Rx feeding/being fed from the OSsynthesis/analysis FB. FIGS. 9C-9F detail the four flavors of sub-bandOFDM Tx/Rx resulting for values of β which are classified according tothe values of β mod 4 (0,1,2,3 respectively in a one-sided modrepresentation). FIGS. 9C,9D correspond to the Tx, detailing the routingfabrics performing zero-padding and circularly rotating the records,detailing the routing fabrics while FIGS. 9E,9F correspond to the Rx,detailing the routing fabrics derotating and dropping a part of therecords.

FIGS. 10A-10C and 11 address the V=4 OS FB based OFDM Tx/Rx in a similarsuccession as shown in FIGS. 9A-9F.

FIG. 12 concludes the OFDM transmission over OS FB structures,establishing that it is not necessary to use the OS FB OFDM Tx shown inFIG. 7B, even if our intent is to use the OS FB OFDM Rx in the RHS ofFIG. 7D. We may instead use a conventional OFDM Tx over the fullchannel. In other words, our OS FB OFDM Rx is compatible with aconventional OFDM Tx as shown in the bottom part of the figure. Thejustification is that the process of FD muxing multiple OFDM signalseach containing N sub-carriers, amounts to generating an extendedbandwidth (M times spectrally wider) OFDM signal containing MNsub-carriers, as graphically indicated. At the Rx, the analysis FBseparates the MN sub-carriers into M groups of N sub-carriers each, witheach such group viewed as a “mini-”OFDM signal in itself, which isdecoded by a sub-band OFDM Rx, one of M such Rx-s, arranged in asub-band Rx array processing all the sub-bands in parallel.

Inspecting FIGS. 6A-6B for the OS FB synthesis-analysis chain and FIGS.7C-7D for the same but in the OFDM case, it is apparent that the V-foldOS synthesis and analysis FBs (the inner two modules in thick dottedlines) are each described as a collection of M BPFs connected to up/downsamplers, hence the hardware complexity of multiple BPFs is very high.The objective now is to find equivalent schemes of reduced hardwarecomplexity for these OS FB modules.

Recall that the FB treated here are of the uniform type, namely all thebandpass filters have identical spectral shape but just differ in theiruniform positioning along the band at intervals R/M=S. The commonalitybetween the filter shapes and the fixed spectral shifts from one filterto the next one suggest that it may be possible to share computationalresources between these filters. Indeed, this is well known in prior artfor CS FBs [,B. Porat, A course in Digital Signal Processing Wiley,1997] which are amenable to highly efficient implementation using afilter array of M polyphase components of the prototype filter as wellas an M-pnt FFT. This implementation is also known in prior art as‘uniform DFT filter bank’ Here we proceed to extend the efficientimplementation from CS FBs to OS FBs.

FIG. 13 discloses our embodiment of hardware-efficient implementationfor the general V=M/L-fold OS Analysis Filter Bank. Here V>1, i.e. M>L,indicating oversampling of the M filter ports. This implementation isbased on an M-pnt IFFT and a discrete-time filter array with impulseresponses determined by the sparse polyphase modulo L components of theprototype filter.

The processing starts with a S/P (1:L splitting here rather than 1:M inthe CS case), then the S/P outputs feed a polyphase filter array but thefilters of the array are now arranged as Single-Input Multiple Output(SIMO) filters. Each such SIMO filter may be viewed as a filter bank of{hacek over (M)} SISO filters with common input. The SISO filtersforming the p-th SIMO filter are all fed by a common output, which isthe p-th output of the S/P. Here the integer {hacek over (M)} is thenumerator obtained by reducing the fraction V=M/L, i.e. {hacek over (M)}is given by the ratio of M and the largest common divisor of M and L.Equivalently, we define {hacek over (M)}, {hacek over (L)} as tworelatively prime integers in the same ratio as M and L:V≡M/L={hacek over(M)}/{hacek over (L)}, e.g. if (M,L)=(16,12), then ({hacek over (M)},{hacek over (L)})=(4,3).

The impulse responses of these SISO filters forming the SIMO filterswill be specified below in terms of the polyphase components of theprototype filter impulse response. The SISO filters may be viewed asorganized in a 2D array of dimension L×{hacek over (M)}. Each of the Lcolumns of this array of filters comprises {hacek over (M)} SISO filtersforming a SIMO filter. We use the index q=0, 1, 2, . . . , {hacek over(M)}−1 as a vertical label for the SISO filters within the p-th SIMOfilter. The (p,q)-th SISO filter is denoted h₀ ^((p,q))[k], labeled interms of its coordinates (p, q) in the 2D filter array. According to theteaching of this invention, the (p, q)-th filter is specified as thesparse double polyphase component of the impulse response h₀[k] of theprototype filter as follows:

k ₀ ^((p,q)) [k]≡h ₀ ^([p]) ^(L) ^([q]) ^(↑{hacek over (M)}) [k]=D ^(q){↑{hacek over (M)}{x[L{hacek over (M)}k+Lq+p]}}.   (2)

Therefore the SISO filters forming the p₀ ^(th)-th SIMO filter are

{h ₀ ^([p) ⁰ ^(]) ^(L) ^([q]) ^(↑{hacek over (M)}) [k]} _(q=0)^({hacek over (M)}−1).

Due to the sparse double polyphase nature of the filters, the totalnumber of non-zero taps in this L×{hacek over (M)} array of sparsefilters equals the number of taps P, of the LPF prototype filter. AllSIMO filters operate at a rate which is a fraction 1/L of the input rateR into the filter bank (equal to the rate at each S/P output), thus thenumber of taps to be evaluated per second is PR/L and the number of tapsto be evaluated per unit input sample is P/L.

The L×{hacek over (M)} outputs of array of filters are organized into{hacek over (M)} rows of L signals each, and each row is presented toone of the {hacek over (M)} ZP&CIRC SHIFT modules, which are verticallystacked above one another in the 3D representation of the figure. The(q₀)-th ZP&CIRC SHIFT module operates on the outputs ρ^([p]) ^(L) [k]

h₀ ^((p,q) ⁰ ⁾[k] of the L filters specified by (p, q₀), p=0, 1, 2, . .. , L−1, according to C_(p) ^([−Lq]) ^(M) {(ρ^([p]) ^(L) [k])

^(ZP[q]:L→M)}, i.e., it zero-pads the L−pnt vector of outputs to lengthM i.e. the index p is extended over the range p=0, 1, 2, . . . , M−1with the subset of values for p=L,L+1, . . . , M−1 set to zero. Then acircular shift with the shift parameter given by [−Lq]_(M) is performedupon the zero-padded M-pnt vector. This completes the description of theZP&CIRC SHIFT modules.

Next, the signals coming out of the ZP&CIRC SHIFT modules (horizontalrows) are organized into vertical columns and the elements of eachcolumn are summed up. The outputs of the M summers are input into anM-pnt IFFT.

We note that each cyclic shifts just represents a fixed permutation ofthe L-input ports into L of the output ports, meaning that each row ofoutputs of the polyphase filters array may be just mapped into apermuted way onto M outputs providing the final vector to be applied tothe M adders. Thus there is no hardware penalty as data does not need tobe shuffled in real-time in actual buffers as in some prior art filterbank implementation. Rather each ZP& CRC SHIFT routing fabric simplyrepresents here a static wiring arrangement of L inputs to M outputsincluding zero-padding of M−L of the outputs.

The following final mathematical formula compactly specifies our highlyefficient implementation of the V=M/L-fold oversampled filter bank:

$\begin{matrix}{{\rho_{\beta}\lbrack k\rbrack} = {\quad^{\beta}{{IDFT}_{p}^{M - {pnt}}\{ {\sum\limits_{q = 0}^{\overset{\Cup}{M} - 1}\; {C_{p}^{{\lbrack{- {Lq}}\rbrack}_{M}}\{ ( {{\rho^{{\lbrack p\rbrack}_{L}}\lbrack k\rbrack} \otimes {h_{0}^{{{\lbrack p\rbrack}_{L}{\lbrack q\rbrack}}_{\uparrow \overset{\Cup}{M}}}\lbrack k\rbrack}} )^{{ZP}:{Larrow M}} \}}} \}}}} & (3)\end{matrix}$

where ρ[k] is the input of the OS analysis FB, ρ_(β)[k] is the β^(th)sub-band output of the OS analysis FB, ρ^([p]) ^(L) [k] is the p-thoutput of the 1:L S/P, {circle around (×)} denotes convolution, and thefilter impulse responses are given by the sparse double polyphases ofthe prototype filter as specified in Eq. (2). Eq. (3) fully specifiesour OS analysis FB efficient processing and is equivalent to the blockdiagram of FIG. 13.

We next consider some specific special cases for various values of theOS parameter, V:

V=M/L=4/3, {hacek over (M)}=4

V=M/L=4, {hacek over (M)}=4

V=M/L=2, {hacek over (M)}=4

FIG. 14 considers an efficient implementation of the SIMO filters forthe first two cases V=4/4=1 and V=4/1=4 for both of which we have {hacekover (M)}=4 , i.e., each SIMO filter maps its input into four outputs.As the composing SISO filter responses are sparse, many of the taps ofthese filters are zero. Although there is no computational impact due tothe presence of the filling zeros, it would nevertheless be more compactin terms of structuring the processing to have a lower number ofnon-sparse filters. The concept according to the teachings of thisinvention is to replace each row of {hacek over (M)} vertically stackedsparse polyphase filters by a single 1 by {hacek over (M)}SIMO-interleaved (SIMO ILV) filter, where the interleaving terminologydescribes how the wires leading to the multiplicative taps areinterleaved as shown.

Although this could be shown in full generality, in FIG. 14 we exemplifythe general case for {hacek over (M)}4=4. The generalization toarbitrary values of {hacek over (M)} is evident. In the implementationshown in figure the null taps are eliminated by using a common delayline (chain of z⁻¹ delays) and tapping the various points along the lineas shown. Multiple lines converging on a node represent here a summationnode. The values of the taps are represented in three different ways,unstructured in terms of the original polyphases modulo L of theprototype filter, or structured as double polyphases mod-L mod-4, orstructured in terms of polyphases mod-4L based on the relation

This figure illustrates the following filters: h₀ ^([Lq+p])^(L{hacek over (M)}) [k]=h_(D) ^([p]) ^(L) ^([q]) ^({hacek over (M)})[k] which for {hacek over (M)}=4 reduces to

h _(D) ^([Lq+p]) ^(4L) [k]=h ₀ ^([p]) ^(L) ^([q]) ⁴ [k].

Now the structure is no longer sparse. There are 16 taps (complexmultipliers shown by little circles) connected to the successive nodesfrom left to right along the delay line. Each delay element output (aswell as the initial delay element input) feeds a taps. The taps fromleft to right are given by the following formulas:

h^([p]) ^(L{hacek over (M)}) [0]=h₀ ^([p]) ^(L) ^([0]) ⁴ [0]=h₀ ^([p])^(L)   [0]

h ₀ ^([L+p]) ^(4L) [0]=h ₀ ^([p]) ^(L) ^([1]) ⁴ [0]=h ₀ ^([p]) ^(L)  [1]

h ₀ ^(2L+p) [0]=h ₀ ^([p]) ^(L) ^([2]) ⁴ [0]=h ₀ ^([p]) ^(L)   [2]

h ₀ ^([3L+p]) ^(4L) [0]=h ₀ ^([p]) ^(L) ^([3]) ⁴ [0]=h ₀ ^([p]) ^(L)  [3]

h ₀ ^([p]) ^(4L) [1]=h ₀ ^([p]) ^(L) ^([0]) ⁴ [1]=h ₀ ^([p]) ^(L)   [4]

h ₀ ^([L+p]) ^(4L) [1]=h ₀ ^([p]) ^(L) ^([1]) ⁴ [1]=h ₀ ^([p]) ^(L)  [5]

h ₀ ^([2L+p]) ^(4L) [1]=h ₀ ^([p]) ^(L) ^([2]) ⁴ [1]=h ₀ ^([p]) ^(L)  [6]

h ₀ ^([3L+p]) ^(4L) [1]=h ₀ ^([p]) ^(L) ^([3]) ⁴ [1]=h ₀ ^([p]) ^(L)  [7]

h₀ ^([p]) ^(4L) [2]=h₀ ^([p]) ^(L) ^([0]) ⁴ [2]=h₀ ^([p]) ^(L)   [8]

h ₀ ^([L+p]) ^(4L) [2]=h ₀ ^([p]) ^(L) ^([1]) ⁴ [2]=h ₀ ^([p]) ^(L)  [9]

h ₀ ^([2L+p]) ^(4L) [2]=h ₀ ^([p]) ^(L) ^([2]) ⁴ [2]=h ₀ ^([p]) ^(L)  [10]

h ₀ ^([3L+p]) ^(4L) [2]=h ₀ ^([p]) ^(L) ^([3]) ⁴ [2]=h ₀ ^([p]) ^(L)  [11]

h₀ ^([p]) ^(4L) [3]=h₀ ^([p]) ^(L) ^([0]) ⁴ [3]=h₀ ^([p]) ^(L)   [12]

h ₀ ^([L+p]) ^(4L) [3]=h ₀ ^([p]) ^(L) ^([1]) ⁴ [3]=h ₀ ^([p]) ^(L)  [13]

h ₀ ^([2L+p]) ^(4L) [3]=h ₀ ^([p]) ^(L) ^([2]) ⁴ [3]=h ₀ ^([p]) ^(L)  [14]

h ₀ ^([3L+p]) ^(4L) [3]=h ₀ ^([p]) ^(L) ^([3]) ⁴ [3]=h ₀ ^([p]) ^(L)  [15]

Each of these formulas contains three equal expressions based onpolyphases mod-4L, double polyphases mod-4 of polyphases mod-2 andpolyphases mod-L.

FIG. 15 considers the reduction of the general efficient analysis FBstructure of FIG. 13 to the special case of an OS analysis FB withV=M/L=4/3, {hacek over (M)}=4. Here the SIMO filters may be realized asin FIG. 14. The figure also specifies the circular shift operators usedafter zero padding, marked onto each of the ZP&CIRC SHIFT modules. Theseshifts are obtained by substituting L=3/4M in the general C_(p) ^([−Lq])^(M) form:

for  V = M/L = 4/3:$C_{p}^{{\lbrack{- {Lq}}\rbrack}_{M}} = {C_{p}^{{\lbrack{{- \frac{3}{4}}{Mq}}\rbrack}_{M}} = \{ \begin{matrix}{{C_{p}^{{\lbrack{{- \frac{3}{4}}{M \cdot 0}}\rbrack}_{M}} = 1};{q = 0}} \\{{C_{p}^{{\lbrack{{- \frac{3}{4}}{M \cdot 1}}\rbrack}_{M}} = C_{p}^{{\lbrack{\frac{1}{4}M}\rbrack}_{M}}};{q = 1}} \\{{C_{p}^{{\lbrack{{- \frac{3}{4}}{M \cdot 2}}\rbrack}_{M}} = C_{p}^{{\lbrack{\frac{1}{2}M}\rbrack}_{M}}};{q = 2}} \\{{C_{p}^{{\lbrack{{- \frac{3}{4}}{M \cdot 3}}\rbrack}_{M}}C_{p}^{{\lbrack{\frac{3}{4}M}\rbrack}_{M}}};{q = 3}}\end{matrix} }$

FIG. 16 describes the routing fabric (wiring of inputs to outputs)corresponding the circular shifts {1, C_(p) ^([1/4M]) ^(M) , C_(p)^([2/4M]) ^(M) , C_(p) ^([3M]) ^(M) } indicated in the last formula, asapplicable to V=4/3-fold OS analysis FB of FIG. 15. The four routingfabrics comprise uniform right cyclic shifts given by {0,1/4M,1/2M,3/4M}applied to the bottom-zero-padded M-points vector obtained from each rowof 3M/4 outputs of the polyphase filter array.

The drawing considers the specific realization of the four ZP&CIRC SHIFTmodules for L=16,M=16, though it is evident how to extend it for anyvalue of M divisible by 4, by partitioning the output ports into foursegments of M/4 ports each, and partitioning the input ports into threesegments of M/4 ports each. The injective mapping of 3M/4 inputs into Moutputs corresponds to the mapping of three segments into four segmentsin the manner shown.

FIG. 17 also addresses the efficient OS analysis FB implementation forthe V=M/L=4/3, {hacek over (M)}=4 case, presenting an alternativeequivalent 2D layout of the 3D layout of FIG. 15, obtained byre-arranging the ZP&CIRC SHIFT and adders.

The processing is partitioned here in terms of polyphases modulo 4 ofthe P/S output ports and likewise polyphases modulo M/4 of the IFFTinput ports. For each value of p modulo M/4 three of the polyphasefilters participate, being fed by three consecutive values of the p-thpolyphase modulo 4. The four respective outputs of each of the SIMOfilters are permuted as shown and are then added up (multiple linesconverging on a node denote addition). The four summing junction outputsare mapped onto four consecutive inputs of the IFFT, with indexesseparated by M/4 as shown.

FIGS. 18A-18C proceed to address the efficient OS analysis FBimplementation for the V=M/L=2, {hacek over (M)}=2, L=M/2 case. Fordefiniteness the specific case of M=16 is further assumed, though thelabeling in the drawing also corresponds to a generic M value. Now, as{hacek over (M)}=2, the SIMO filters become Single Input Dual Output(SIDO) ones, and there are just two ZP&CIRC SHIFT modules, with circularshifts given by

$C_{p}^{{\lbrack{- {Lq}}\rbrack}_{M}} = {C_{p}^{{\lbrack{{- \frac{1}{2}}{Mq}}\rbrack}_{M}} = \{ \begin{matrix}{{C_{p}^{{\lbrack{{- \frac{1}{2}}{M \cdot 0}}\rbrack}_{M}} = 1};{q = 0}} \\{{C_{p}^{{\lbrack{{- \frac{1}{2}}{M \cdot 1}}\rbrack}_{M}} = C_{p}^{{\lbrack{\frac{1}{2}M}\rbrack}_{M}}};{q = 1}}\end{matrix} }$

Thus, as shown in FIG. 18B the top ZP&CIRC SHIFT module comprises justthe zero padding copying the M/2 inputs into the first M/2 outputs andadding M/2 zeros underneath, whereas the bottom one corresponds tomapping the M/2 inputs into the lower half of the output M points, withthe top half consisting of zeros.

Notice that the M/2 non-zero elements in the two M-pnt outputs of thetwo ZP&CIRC SHIFT modules do not overlap in index. This means that the2-pnt adders are always presented with a single non-zero input, whilethe other input is zero, thus the adders may be discarded and thenon-zero inputs which were supposed to go into the adders may bedirectly routed to the input of the IFFTs where the adder was supposedto be connected. This implies that the ZP&CIRC SHIFT modules as well asthe adders may be discarded. Instead, as shown in FIG. 18C, the upperoutputs (labeled q=0) collected from all the L=M/2 SIDO filters are thendirectly mapped onto the top half of M/2 inputs of the IFFT, while thebottom outputs (labeled q=1) of the L=M/2 SIDO filters are directlymapped onto the bottom half of M/2 inputs into the IFFT.

FIGS. 19A-19C present a planar block diagram layout equivalent to the 3Ddiagram of FIGS. 18A-18C. FIG. 19A is based on L=M/2 SIDO-ILV filterswith inputs fed by the L=M/2 outputs of the S/P, and the pair of outputsof the p-th filter are connected to the p-th and (p+M/2)-th input portsof the IFFT. FIGS. 19B-19C detail two alternative equivalentimplementations of the SIDO filters.

FIG. 20 shows an alternative embodiment of the efficient OS analysis FBimplementation for the V=M/L=2, {hacek over (M)}=2, L=M/2 case. Thisembodiment is based on an array of M filters with impulse responsesgiven by the sparse double polyphase components modulo 2 of thepolyphase components modulo M/2 of the prototype filter impulseresponse, the definition of which is repeated here:

Even/Odd (sparse) polyphases of the polyphase components modulo M/2 ofthe prototype filter h₀[k]:

h₀ ^([p]) ^(M/2) ^([q]) ² [K]≡h₀[1/2M(2k+q)+p]

h₀ ^([p]) ^(M/2) ^([q]) ^(↑2) [k]≡D^(q){↑2{h₀[1/2M(2k+q)+p]}}

This figure may be derived from FIG. 19C viewing each SIDO filter as apair of SISO filters in parallel (each given by the corresponding sparsedouble polyphase components), then modifying the resulting block diagramby moving the pair of SISO filters without altering the topology, say,having the first SISO filter of the p-th SIDO filter drawn opposite tothe p-th input to the IFFT while the second SISO filter of the p-th SIDOfilter is drawn opposite to the (p+M/2)-th IFFT input. The two blockdiagrams of FIGS. 19A and 20 have the same computational complexity,though FIG. 19A with SIDO filters given by 19B may be more compact torealize as it avoids the zero taps and shares the delay lines across thetwo SISO filters forming each SIDO filter.

FIGS. 21,22 proceed to address the efficient OS analysis FBimplementation for the V=M/L=4, {hacek over (M)}=4, L=M/4 case, startingwith specifying in FIG. 21 the routing fiber corresponding V=4. Fordefiniteness the specific case of M=16 is further assumed, though thelabeling in the drawing also corresponds to a generic M value.

As the value of {hacek over (M)}=4 is identical to that obtained for thepreviously treated V=4/3 case, a routing fabric structure similar tothat of FIG. 16 will be obtained in FIG. 21, also featuring 1 by 4 SIMOfilters as well as four ZP&CIRC SHIFT modules though now the number ofSIMO filters and the number of inputs into each ZP&CIRC SHIFT module isM/4 rather than 3M/4. The SIMO filters are identically wired internally,i.e. the diagram of FIG. 14 equally applies to the two cases V=4/3 andV=4, however as the values of L are different (L=3M/4 vs. L=M/4), thepolyphases from which the taps are derived are also different, hence theresulting taps values are different. The respective sets of taps in thetwo respective cases are obtained by substituting L=3M/4 vs. L=M/4 inthe tap formulas which were specified above for FIG. 14. The currentFIG. 21 describes the structure of the four ZP&CIRC SHIFT modules forthe current V=4 case, which now differ from the those shown in FIG. 16for the previous V=4/3 case. Now there are evidently just L=M/4 non-zeroinputs, and the applicable circular shifts are now given by:

for  V = M/L = 4:$C_{p}^{{\lbrack{- {Lq}}\rbrack}_{M}} = {C_{p}^{{\lbrack{{- \frac{1}{4}}{Mq}}\rbrack}_{M}} = \{ \begin{matrix}{{C_{p}^{{\lbrack{{- \frac{1}{4}}{M \cdot 0}}\rbrack}_{M}} = 1};{q = 0}} \\{{C_{p}^{{\lbrack{{- \frac{1}{4}}{M \cdot 1}}\rbrack}_{M}} = C_{p}^{{\lbrack{\frac{3}{4}M}\rbrack}_{M}}};{q = 1}} \\{{C_{p}^{{\lbrack{{- \frac{1}{4}}{M \cdot 2}}\rbrack}_{M}} = C_{p}^{{\lbrack{\frac{1}{2}M}\rbrack}_{M}}};{q = 2}} \\{{C_{p}^{{\lbrack{{- \frac{1}{4}}{M \cdot 3}}\rbrack}_{M}}C_{p}^{{\lbrack{\frac{1}{4}M}\rbrack}_{M}}};{q = 3}}\end{matrix} }$

These circular shifts are implemented by the wiring diagrams of thefigure, shown for the particular value M=16, though the wiring isgeneralized in an evident way for any value of M which is a multiple of4.

FIG. 22 reduces the general 3D diagram of FIG. 13 for the V-fold OSanalysis FB to the specific case that V=4, taking advantage of thespecific structure of the ZP&CIRC SHIFT outputs as was shown in FIG. 21.One key observation evident from inspecting FIG. 21 is that that the M/4non-zero elements in the four M-pnt outputs of the four ZP&CIRC SHIFTmodules do not overlap in their indexes. This means that the 4-pntadders of the generic block diagram of FIG. 13 are always presented witha single non-zero input, while the other three inputs are zero, thus theadders may be discarded and the non-zero inputs which were supposed togo into the adders may be directly routed to the inputs of the IFFTs,whereat the adders were supposed to be connected. This implies that theZP&CIRC SHIFT modules as well as the adders may be discarded. Instead,as shown in the current FIG. 22, the top row of outputs (labeled q=0) ofthe L=M/4 SIDO filters are then directly mapped onto the top quarter ofM/4 inputs into the IFFT, the second row of outputs (labeled q=1) of theL=M/4 SIDO filters are then directly mapped onto the second quarter ofM/4 inputs into the IFFT, the third row of outputs (labeled q=2) of theL=M/4 SIDO filters are then directly mapped onto the third quarter ofM/4 inputs into the IFFT and finally the bottom row of outputs (labeledq=3) of the L=M/4 SIDO filters are then directly mapped onto the bottomquarter of M/4 inputs into the IFFT. Thus, this scheme discards theZP&CIRC SHIFT altogether, instead mapping the M outputs of the 2D arrayof the sparse double polyphase filters directly onto the M inputs of theIFFT, row by row into a corresponding quarter segment of the IFFT inputsas shown.

FIGS. 23A-23B, 24A-24B and 25A-25B show yet another set of embodimentsfor the OS analysis FB for V=2,4/3, 4 respectively. The idea here is touse the CS FB known prior art building blocks in order to realize thenovel OS FB structures, by combining multiple CS FBs with additional‘glue’ elements in order to generate the new OS FBs. Such an approach,re-using the simpler CS FBs as building blocks, allows for structuredhardware design, as the CS FB is a well-defined element, which is justrepeated several times. Moreover, hardware cores are made available byFPGA vendors, enabling rapid prototyping of OS FB in terms of the moreelementary CS FB building blocks. It should be mentioned that all the Mfilters impulse responses in these drawings correspond to the Mpolyphases modulo M of the prototype filter, although we do not use thenotation for polyphases as used throughout this application, but welabel these filter impulse responses in these figure by an evidentsimplified shorthand notation.

FIGS. 23A-23B describes the alternative construction of the twice OS(V=2) analysis FB based on a pair of interconnected identical M-pnt CSFBs. In both figures the input signal into the FB is split into twocopies, one of which is delayed by M/2 samples (half the number ofoutput ports of the FB) relative to the other. The two copies are inputinto two identical CS analysis filter banks, realized efficiently as perprior art based on a 1:M S/P an array M filters with impulse responsesgiven by polyphase components modulo M of the prototype filter, and anM-pnt IFFT, as described in FIG. 12. In FIG. 23A the M outputs of the CSanalysis FB, to which the input signal was not delayed, are modulated byalternating (−1)^(β) factors, where β=0, 1, 2, . . . , M−1 is the outputport index of the IFFT. Therefore, the sign of every other IFFT outputis flipped. The corresponding elements of resulting M-pnt vectors arethen paired up (the β^(th) element of the first vector is paired up withthe β^(th) element of the second vector) and the pairs are input into2:1 P/S modules, such that the β^(th) such module generates the β_(th)FB output ρ_(β)[k]. In FIG. 23B the alternating (−1)^(β) modulation isreplaced by a half-band circular shift of the inputs to the IFFT, asdescribed by the C_(p) ^([M/2]) ^(M) operator, in effect interchangingthe first M/2 and the second M/2 inputs, as depicted in the crossedwires diagram. The rest of the processing is like in FIG. 23A.

In terms of operating rates, starting at rate R, the rate of arrivals ofM-pnt blocks is R/M, after the S/P 1:M operation the blocks appear inparallel at the M outputs of the S/P; this rate is maintained throughoutthe filtering and the IFFT, which produces parallel blocks of M samplesat the same rate of R/M per second. This is also the rate at whichsamples appear in each of the output ports of the IFFTs. Every two portsof corresponding index across the four IFFTs are collected into aparallel pair of signals which are time-division multiplexed by the 2:1P/S modules, bringing the rate up by a factor of 2 to a 2R/M output rateper output wire (cf. the rate R/M per wire out of an M-pnt CS analysisFB—here the rate is V=2 times faster).

FIGS. 24A-24B describe the alternative construction of the V=4/3 OSanalysis FB based on four interconnected identical M-pnt CS FBs. Fourcopies of the input signal, relatively delayed in steps of L=3M/4, aregenerated as shown using three delay elements of L=3M/4 samples each.Each of these four signals is input into a modified CS analysis FB,which has been altered as follows:

-   (i): The conventional 1:M S/P is replaced by a 1:3M S/P, at the    output of which just the top M inputs are retained. This corresponds    to slowing down the processing rate following this S/P by a factor    of 3, as (input into the polyphase filter array), since the next 2M    samples following the retained M samples are discarded, then a new    set of M samples is retained and passed to the polyphase filter    array, etc., i.e. we process M samples every 3M samples which    corresponds to 3-fold downsampling of the input rate. Notice that    this is not equivalent to down-sampling by a factor of 3 followed by    a 1:M S/P, nor to a 1:M S/P followed by down-sampling by a factor of    3, but this is a different kind of block-oriented down-sampling,    where we retain one block out of each 3 consecutive input blocks.-   (ii): The CS FBs are modified to include circular shifts at the IFFT    inputs amounting to a permutation of the filters outputs by means of    the circular shifters, prior to their applications to the IFFTs. The    wiring diagrams of the circular shifters are shown in FIG. 24B.    Alternatively, not illustrated, the circular shifts in FIG. 24A may    be removed at the IFFT inputs and rather implemented as modulations    by the respective phase rotator factors 1^(β), (−j)^(β)(−1)^(β),    j^(β) applied at the outputs of each of the four IFFTs. Thus, the    four structures combined in FIG. 24A to yield the overall OS    analysis FB are not strictly CS analysis FBs, but are very close in    the sense that they use the same polyphase filter components modulo    M, and the same M-pnt IFFT size.

The corresponding elements of four resulting M-pnt vectors at the fourmodified CS analysis FB outputs are then organized in quadruples (theβ^(th) elements of each of the four vectors are paired up together) andthe resulting quads of signals are input into 4:1 P/S modules (M ofthem), such that the β^(th) such module generates the β^(th) FB outputρ_(β)[k].

In terms of operating rates, starting at input rate R, the rate ofarrivals of blocks of M points is R/M, after the S/P 1:3M operation therate of retained blocks is R/(3M), this rate is maintained throughoutthe filtering and the IFFT, which produces parallel blocks of M samplesat the same rate of R/(3M) per second. This is also the rate at whichsamples appear in each of the output ports of the IFFTs. Every fourports of corresponding index across the four IFFTs are collected into aparallel quad of signals which are time-division multiplexed by the 4:1P/S modules, bringing the rate up by a factor of 4 to a 4R/(3M) outputrate per output wire (cf. the rate R/M per wire out of an M-pnt CSanalysis FB—here the rate is V=4/3 times faster).

FIGS. 25A-25B describes the alternative construction of the V=4 OSanalysis FB based on four interconnected identical M-pnt CS FBs. Fourcopies of the input signal, relatively delayed in steps of L=M/4, aregenerated as shown using three delay elements of L=M/4 samples each.Each of these four signals is input into a modified CS analysis FB whichincorporates circular shifts ahead of the IFFTs as shown on the right ofthe figure.

In terms of operating rates, starting at rate R, the rate of arrivals ofM-pnt blocks is R/M, after the S/P 1:M operation the blocks appear inparallel at the M outputs of the S/P; this rate is maintained throughoutthe filtering and the IFFT, which produces parallel blocks of M samplesat the same rate of R/M per second. This is also the rate at whichsamples appear in each of the output ports of the IFFTs. Every fourports of corresponding index across the four IFFTs are collected into aparallel quad of signals which are time-division multiplexed by the 4:1P/S modules, bringing the rate up by a factor of 3 to a 4R/M output rateper output wire (cf. the rate R/M per wire out of an M-pnt CS analysisFB—here the rate is V=4 times faster).

FIGS. 26, 27A-27B, 28, 29A-29B, 30 and 31 describe various embodimentsof the OS synthesis FBs. In fact for every embodiment we heretoforederived for OS analysis FB we may readily generate a correspondingembodiment of an OS synthesis FB by using a signal processing dualityproperty which states the following: by exchanging adders-splitters,P/S-S/P, FFT-IFFT, SIMO-MISO and by reversing the order (labeling) ofFIR filter taps (time reversing the tap sequences), a meaningful dualcircuit generating the inverse function of the original circuit isobtained. In particular notice that multiple arrows coming in into ajunction denote summations of the signals on the incoming arrows, whichis the dual of multiple arrows coming out of a junction indicating asplitter.

While in the OS analysis FB diagrams shown heretofore the signal flowhas always been from left-to-right, in these dual synthesis FB figures,the signal flow is reversed, proceeding from right-to-left and applyingthe duality mapping of elements. Thus, the duality property amounts totime-reversal—reversing the direction of flow through the signalprocessing structures.

FIG. 26—the duality property is applied to a general V-fold analysisfilter bank of FIG. 13 in order to generate the hardware-efficientgeneric V-fold synthesis filter bank of the current figure.

FIGS. 27A-27B presents the embodiment of a hardware-efficient 2×OSsynthesis filter bank in terms of Dual Input Single Output (DISO)filters. This figure may be either obtained as a special case of thegeneral V-fold synthesis, or readily derived by duality from the 2×OSanalysis FB structure of FIG. 19A. The SIDO filters of FIG. 19A arereplaced here by DISO filters, the IFFT of FIG. 19A is replaced by anFFT, the S/P there replaced here by a P/S, etc. In FIG. 23B we detailthe DISO filters. This figure is the dual of FIG. 19B.

FIG. 28 presents an alternative embodiment of a hardware-efficient 2×OSsynthesis filter bank in terms of a pair of CS synthesis FBs. Thisfigure is the dual of the 2×OS analysis FB realization of FIG. 23B.

Figs. A-29,B present the embodiment of a hardware-efficient V=4/3 OSsynthesis filter bank. The duality with respect to thehardware-efficient V=4/3 OS analysis filter bank of FIG. 24A-24B isevident. In terms of processing rates, each input sub-band is presentedat OS rate

$\frac{4}{3}\frac{R}{M}$

rather than the CS rate of

$\frac{R}{M}$

The TDM demuxing means that each of the four inputs out of the 1:4 S/Phas ¼ of the rate i.e.

${\frac{1}{4}( {\frac{4}{3}\frac{R}{M}} )} = {\frac{1}{3}{\frac{R}{M}.}}$

This is the rate at which blocks are collected across the FFT inputs,and also the rate of the filtering array, thus M-pnt blocks appear ateach filtering array outputs at this rate. The 3M:1 P/S TD muxing actionraises the sampling rate by a factor of 3M, yielding the output rate

${3\; {M( {\frac{1}{3}\frac{R}{M}} )}} = {R.}$

The four P/S outputs are superposed at this rate yielding the finaloutput rate R.

FIGS. 30 and 31 present the embodiment of a hardware-efficient V=4 OSsynthesis filter bank. The duality with respect to thehardware-efficient V=4 OS analysis filter bank of FIGS. 25A-25B isevident.

This completes the disclosure of oversampled filter bank processingstructures. The OS FB modules are next used as building blocks form morecomplex systems for optical transmission, namely the optically coherenttransmitters (Tx) and (Rx), based on OFDM and single-carrier (SC)modulation formats. Overviewing these following embodiments, OFDM Rx-susing oversampled filter banks will be referred to as Multi-Sub-BandOFDM Rx-s. We shall also introduce an OS FB version of the prior-artDFT-spread (DFT-S) OFDM [DFT-Spread OFDM for Fiber NonlinearityMitigation Yan Tang, William Shieh, and Brian S. Krongold, IEEEPHOTONICS TECHNOLOGY LETTERS, VOL. 22, pp. 1250-1252, 2010] referred toas MSB DFT-S OFDM. We also introduce Nyquist shaped single carrier Txand corresponding OS FB Rx embodiments referred to as SC-MSBE as well asa new class of embodiments with variable numbers of sub-single-carriersper channel, using sub-band bonding, referred to as bonded-MSBE.

In this application there is also provided Parallel-to-Parallel P/Pmodule specified by the ratio p:q where p,q are two integers one ofwhich divides the other. The P/P module is specified to take p parallellines and produces q parallel lines. E.g. when p>q and q divides p, i.e.we ‘down-parallelize’ then you can realize the P/P as the paralleljuxtaposition of q P/S modules, each being (q/p):1 sized. Notice thatP/P with p:p, i.e. p=q is just a trivial identity system taking pparallel lines as input and presenting the same p parallel lines asoutput. We shall resort to the p:p P/P module (an identity system) as agraphical means to compactly depict a multiple wires conduit. Many ofthe block diagrams of this disclosure comprise parallel to serial andserial to parallel converters. These conversion modules may actually beused in their strict defining sense, as conceptual representation ofdata flow, however in practice the nominal serial inputs or outputs tothese P/S and S/P conversion modules are often not serialized at all butare handled in parallel in order to ease the high speed processing.Thus, the p:1 P/S modules are often implemented as p:q P/P modules andthe 1:p S/P modules are often implemented as q:p P/P with the q numberof parallel ports corresponding to the nominal serial input or outputpossibly differing from p, selected according to hardware processingconvenience. Thus, whenever we mention S/P and P/S in the claims itshould be understood that these models may be implemented in practice asP/P.

FIG. 32,33 present our MSB OFDM PDM Rx preferred embodiment as per theteaching of this invention. At the top level view of the Rx (FIG. 32)each of the complex-valued signals for each of the two X and Ypolarizations is passed through a 1:M 2×OS analysis filter bank toseparate the sub-bands to M distinct output ports. In our exemplarysystem the total channel bandwidth is B=25 GHz, and M=16, i.e. eachpolarization signal is FD demultiplexed by the FB into 16 sub-bands. Theextreme sub-band which is split between the two ends of the spectrum, isdedicated to an ADC anti-aliasing filter guard-band, whereas theremaining 15 sub-bands fill up the channel information bandwidth, takenhere in this exemplary system as 25 GHz. The exemplary ADC sampling rateis then

${R = {{\frac{M}{M - 1}B} = {{\frac{16}{15}( {25\mspace{14mu} {GHz}} )} = {26.6\mspace{14mu} {GS}\text{/}s}}}},$

allowing to accommodate the ADC anti-alias guardband over the 16thsub-band, and leaving 15 sub-bands for processing and detecting the netdata modulated over the 25 GHz, i.e., each sub-band has a spectral widthof 1/15(25GHz)=1.66 GHz. The sub-bands with corresponding indexes fromthe X and Y FBs are paired up to feed identical sub-band Rx-s, in turndetailed in FIG. 33. The QAM-demapped X and Y digital output pairs ofeach of the sub-band Rx-s, are and data-multiplexed to generate theoverall DATA OUT bitstream.

Adopting a bi-polar indexing convention for the M−1 sub-bands (and notcounting the extreme ADC-transition sub-band), the i-th X-sub-band andi-th Y-sub-band outputs of the two FBs are routed to feed the i-th Rx ofthe sub-band processor array, for i=−7,−6, . . . , −1, +1, . . . , +7.Notice that the i=0 index is skipped, since in our preferred embodimentwe dedicate this mid-sub-band to pilot tone transmission, hence thissub-band is not routed to one of the sub-band Rx-s of the sub-bandprocessor array but is routed instead to a mid-sub-band processor, usedto extract the transmitted pilot in order to assist in channelestimation and linear and nonlinear phase noise compensation. It is notnecessary however to adopt this pilot sub-band strategy, dedicating awhole sub-band to the pilot signal, however in our preferred embodimentwe do so.

The sub-band processing associated with the MSB OFDM Rx of FIG. 32 isshown in FIG. 33, which details the sub-band OFDM Rx. As our analysisFBs are 2×OS (V=2), the twice overampling implies that for the 1.66 GHzsub-bands the sampling rate at each of the 15 FB output ports will be2×1.66=3.3 GS/s. Following a single polarization, say the X one, the SBsignal is input to the IQI COMP module used for compensating the IQimbalance. This module is further described in FIG. 43A-43F. Then thesignal undergoes CFO and (NL) PN demodulation, removing frequency offset(deviation of the frequency of the laser LO relative to the incomingoptical signal center frequency) and also removing some of the linearand non-linear phase noise. Next, the signal is passed through the CTOrecovery, applying a suitable integer delay in order to temporally alignthe signal with the beginning of the valid FFT window for OFDM. Theimpairment parameters IQI, CFO, CTO and also possibly SFO, to be appliedto the chain of compensators, are derived from a Joint IQI, CFO, CTO,SFO estimation module, the internals of which are detailed inFIG.43A-43F.

Notice that the uniformly spaced frequency offsets of the successivesub-bands induce, via the CD, a group delay proportional to the centerfrequency of each sub-band, such that the various sub-bands come outtime-misaligned with timing offsets forming an arithmetic sequence. Byfitting a straight line to the estimated coarse timing offsets, it ispossible to estimate the amount of CD, providing an efficient CDmonitoring means, however unlike in a conventional coherent Rx whereinCD monitoring and estimation is a very essential function, here explicitCD estimation is not required at all for Rx operation here (inconventional receivers where where CD monitoring and estimation isindispensable and it is necessary to supply a separate means to estimateCD in the channel in order to correctly set the coefficients of the CDequalizer).

After dropping the CP, which is quite short here—in our exemplary systemusing N=64 sub-carriers per sub-band (represented by 128 samples after2×OS by the FB), the CP is just one sample, i.e. 1:128=0.78% overhead),the signal is conditioned for performing the OFDM FFT analysis in orderto extract the sub-carrier complex amplitudes. The reason we are ablekeep the CP very short in our scheme (thus attain high spectralefficiency) is that the CD distortion causing loss of orthogonality ofthe sub-carriers, need only be considered over the restricted bandwidthof each sub-band. Once the filter bank separates out the sub-bands andprocesses each one of them separately, each sub-band is an effectivenarrow-band OFDM system in itself. Thus the delay spread induced by CDis very small per sub-band. This is one key advantage of MSB OFDM whichachieves in effect reduced guard interval (reduced CP) withoutnecessitating the heavy FDE pre-FFT channel equalizer required inoptical long-haul implementations of OFDM transmission in order to keepthe CP relatively short. Such pre-FFT equalizer is eliminated in our MSBOFDM Rx.

Continuing along the processing chain in FIG. 33 (which amounts to anexpanded version of the sub-band OFDM Rx shown earlier in FIG. 8), thesize of the FFT is 2N=128 points, and the FFT is followed by a 2-folddecimation step, performed by the EXTRACT HIGH/LOW HALF-BAND module,dropping one half of the FFT output points, retaining just N=64 points.This 2:1 step is part of the oversampled FB operation, as described inFIG. 8B, where we have seen that the dropped half-band alternatesbetween the high and low half-bands: the high half-band is dropped forthe even sub-band indexes and the low half-band is dropped for the oddsub-band indexes. At this point the rate has been halved, in ourexemplary system down to 3.2 GS/s/2=1.66 GS/s per sub-band (we recallthat each sub-band has bandwidth of S=1.66 GHZ so now we are nowsampling at the Nyquist rate for each sub-band).

Next, as the received X and Y signals are linear combinations of thetransmitted X and Y polarizations, 2×2 MIMO equalization is performedper sub-carrier. Each signal pair formed by n-th sub-carrier from the Xpolarization and n-th sub-carrier from the Y polarization for n=0, 1, .. . , N−1, is passed through a 2×2 MIMO transformation consisting of abutterfly of four complex multipliers as shown. The four complex tapvalues are set by a polarization adaptive tracking module.

Finally, the POL-cross-talk free sub-carriers for the X and Y POLs,emerging from the array of 2×2 MIMO EQZs, are presented in parallel totwo Carrier Recovery modules one for each polarization. In the preferredembodiment we would use Multi-Symbol Delay Detection (MSDD) for theCarrier Recovery method [see our other patent application], howeverother methods may be used to estimate and mitigate the residual phasenoise in each POL of each sub-band prior to decision (decision, i.e.slicing, is performed at the back end of each CR module).

The outputs of the CR+decision modules in the sub-band processor arrayrepresent the decided symbols for each of the X and Y polarizations ofeach of the sub-bands. Demapping each of the QAM sub-carrier symbols andmultiplexing all resulting bit-streams yields the total bit-streamrepresenting the equivalent of the 2×25 GBd signal.

Recapping the overall operation and rationale of the sub-band Rx, theprocessing is significantly improved, in terms of complexity as well asperformance, by the fact that each sub-band is narrowband, hence it isfrequency-flat. Therefore no memory (no FIR filtering) is required inthe MIMO equalization, which may be accomplished by a simple 2×2 matrixconsisting of four complex multipliers. Moreover, the sole effect of CDwithin the spectral extent of each sub-band is essentially a pure delay(corresponding to the group delay of the sub-band at its centerfrequency), while the distortion due to CD is negligible, provided thatthe sub-band is sufficiently spectrally narrow (in our exemplary system,the 1.66 GHz bandwidth per sub-band satisfies this condition—as ameasure of the negligible CD distortion, the number of taps necessary toequalize CD is less than a single tap for 2000 Km standard fiber over1.66 GHz spectral window).

Thus, the CTO compensation is simply performed by an integer delay equalto the group delay of the sub-band rounded off to a multiple of thesampling interval. The estimation of CTO is more robust for thenarrowband sub-bands than for a conventional OFDM receiver operatingbroadband over the full channel. Such CTO estimation may be carried outby the Schmidl-Cox algorithm or by the Minn algorithm, both used inwireless communication, as further detailed below in FIG. 43 addressingthe impairment parameters estimation aspects. The Minn algorithm has notbeen used before in optical communication, but the fact that we apply itper sub-band facilitates its advantageous usage. As for the residual FTO(fine timing offset less than one sample interval), its compensation inthe OFDM Rx is well known to amount to one-tap equalization (in thescalar case). In our dual-polarization vectorial case, the four complexmultipliers would automatically assume a common value automaticallycompensating of FTO compensation, under the control of the adaptivetracking algorithm driven by a training sequence or decision directed.It follows that FTO estimation is not even necessary in our MSB OFDM Rx.There is no need to estimate FTO separately, as the POL ADAPTIVETRACKING module which is training sequence based (e.g. using the LMSalgorithm, or for QAM a decision-radius directed algorithm)automatically adjusts the linear phase resulting from the FTO, lumpingit with the per sub-carrier phase deviation resulting from otherimpairments, such as residual CD or any residual frequency responsealong the channel or in the electro-optic Tx BE or in the Rx FE.

This completes the description of the MSB OFDM sub-band Rx. The overallRx processing is seen to consist of a “divide&conquer” approach, withthe top layer comprising the two filter banks (one per POL) whichperform the “divide” of the wide channel spectrum into multiplenarrowband sub-bands. The “conquer” occurs in the sub-band Rx processorarray, as the sub-bands are easier to contend with individually, ratherthan processing the overall full channel at once.

In fiber communication, the CD-induced impairments grow quadratically inbandwidth, thus reducing bandwidth by a factor of M yields a beneficialCD-impairment reduction by a factor of M². The narrow-bandfrequency-flat sub-bands also imply that adaptive equalization algorithmare going to better convergence and attain faster convergence rates.Therefore, basing the optical coherent OFDM Rx on the OS analysis filterbanks is very beneficial in terms of performance It may be shown bycounting the heaviest processing elements, namely the multipliers, thatthe OS analysis FB Rx of this figure has a substantial complexityreduction advantage over conventional prior art OFDM Rx.

Novel MSB DFT-S OFDM

FIG. 34 presents the sub-band DFT-S Rx associated with our MSB DFT-SOFDM PDM Rx, as per the teaching of this invention. Notice that the toplevel of the MSB-DFT-S OFDM Rx is identical to the top level of the MSBOFDM Rx (without DFT-S) as shown in FIG. 32, which doubles up as bothMSB OFDM Rx and MSB DFT-S OFDM Rx, with the two versions differing inthe sub-band level OFDM Rx-s.

Our MSB DFT-S OFDM PDM Rx is driven by a prior art DFT-S PDM OFDMtransmitter. Both our MSB OFDM and MSB DFT-S OFDM embodiments use thesame top level Rx structure as shown FIG. 32, the difference between thetwo embodiments being in the respective sub-band Rx-s in FIGS. 33 and34. In fact the only difference between our sub-band Rx embodiments(without and with DFT-S) is only in the incorporation of the N-pnt IFFTsused for DFT-despreading (one for X and one for Y) in the MSB DFT-S OFDMsub-band Rx of FIG. 34, at the outputs of the POL-demux modules. Thespectral handling of the overall MSB DFT-S OFDM Rx is generallyidentical to a conventional prior-art DFT-S OFDM Rx, in the sense thatmultiple sub-single-carriers (SSC) each coinciding with a sub-band,which are multiplexed by a conventional prior art DFT-S OFDM Tx with itsnumber of bands equal to the number of sub-bands in the OS analysis FBbased MSB Rx, are each demultiplexed by the MSB Rx shown in FIG. 32 andthe sub-bands are processed in parallel by the sub-band DFT-S OFDM Rx-sof FIG. 34. The realization complexity of our MSB DFT-S OFDM Rx is lowerthan that of the prior art DFT-S OFDM Rx.

FIGS. 35,36 present alternative embodiments of our Nyquist-ShapedSingle-Carrier (NS-SC) PDM Tx, as per the teaching of this invention.The term Nyquist-Shaped refers to a channel spectrum which isnear-rectangular, enabling highly spectrally efficient packing ofmultiple channels tightly in frequency, in a coherent WDM system.Moreover, the transmitted signal after suitable reception should not inprinciple exhibit ISI. Furthermore, the ICI between adjacent NS-SCsignals forming a WDM multiplex should also be zero in principle. Theseconditions are only approximately satisfied in practice, yet nominallythe amounts of ISI and ICI would be lowered in the Nyquist shapeddesign.

The NS-SC Tx is innovatively realized here as a special case of DFT-SOFDM in the unconventional limiting case of having just onesub-single-carrier, with a few adjustments in terms of the sub-carriersmapping, comprising an innovative method of pilot sub-band insertion. Asshown in FIG. 35, the disclosed Tx back-to-back connects a single K⁻-pntFFT with a K₊-pnt IFFT. Here K⁻<K₊. Unlike in prior art DFT-S OFDM, herehowever there is a single DFT-spreading FFT, rather than multiple ones.In addition, also per the teaching of this invention, the method shownfor pilot sub-band insertion is to insert a gap in the mapping of theoutputs of the K⁻-pnt FFT onto the inputs of the K₊-pnt IFFT. This gapprovides the pilot sub-band, within which the training signal consistingof one or more pilots or other band-limited training sequences may beinserted. By a similar technique, other gaps may be introduced,providing other training or service channels. Notice that there areadditional inputs at the extremities of the IFFT input record which areset to null (zero-padded) in order to provide anti-aliasing filterguardbands for the DACs. These gaps are actually symmetrically appliedwith K_(ENDS)/2 inputs nulled out at either end. As the signal generatedby this transmitter is intended to be used with a sub-banded based Rx,it is worth setting all spectral widths (counted as integer sub-carriernumbers) equal to multiples of the sub-band spectral width, which equalsN. In addition, it would be convenient to make the ratio K⁻/K₊<1 asclose to unity as possible, i.e. the total guardband K₀+K_(ENDS) besmall. Moreover, it is convenient to select one of the (I)FFTs to have asize which is a power-of-two, for ready implementation according to theCooley-Tuckey (I)FFT algorithm (as K⁻/K₊≈1 it is not possible to haveboth K⁻, K₊ be powers-of-two, but we should strive to make just one ofthem be a power-of-two, while letting the other one consist of apower-of-two times a small prime number different than 2 such as 3,5 or7. All these requirements are reconciled for the following exemplarydesigns (all with N=64) wherein M is the raw sub-bands number, N is thenumber of sub-carriers per sub-band

K ⁻=(M−2)N=14N=896, K ₊ =MN=16N=1024, K ₀ =N=64=K _(ENDS) , M=16

K ⁻=(M−2)N=16N=1024; K ₊ =MN=18N=1152, K ₀ =N=64=K _(ENDS) , M=18

K ⁻=(M−4)·N=16N=1024; K ₊ =MN=20N=1280, K ₀=2N=128=K _(ENDS) , M=20

In the following figures we assume the first exemplary design with896-pnt DFT-spread FFT feeding into the 1024-pnt FFT. This designsapparently has somewhat lower complexity than the other two designs. Thepilot and the end intervals are then precisely one sub-band intervalwide. This Tx is compatible with SC-MSBE Rx of FIGS. 37A-37B below.

Notice the permuted mapping of output ports of the FFT into the inputports of the IFFT, including the insertion of the pilot, as well as thealternating sign modulation by the (−1)^(k), all intended to worktogether such that a time-domain input of a certain frequency at the FFTinput is mapped into the same time-domain output (albeit upsampled) atthe DAC inputs.

FIG. 36 presents an alternative embodiment of our Nyquist-ShapedSingle-Carrier (NS-SC) PDM Tx, as per the teaching of this invention,differing from that FIG. 35 in the different treatment of the mapping ofoutput ports of the FFT into the input ports of the IFFT, including theinsertion of the pilot. Now, the modulation by (−1)^(k) of FIG. 35 iseliminated and the two FFT halves of the output are not crossed over,however the pilot sub-band must now be inserted at the edge inputs ofthe IFFT, whereas the DAC transition spectral region corresponds toleaving a gap in the center IFFT inputs as shown.

FIGS. 37A-37B disclose an embodiment of a single-carrier (SC) Rx usingour novel multi-sub-band (MSB) processing. This receiver is referred tohere as SC-MSBE Rx, where MSBE stands for MSB Equalization. According tothe teachings of this invention, the Rx may operate with the singlecarrier spectrally sharp Tx of FIGS. 35,36 or with a conventionalsingle-carrier Tx (provided certain measures are taken in the prior artTx to provide training sequences as required for the initialization andchannel estimation of our disclosed Rx).

The Rx disclosed in FIG. 37A is based on Single-CarrierMulti-Sub-Band-Equalization (SC-MSBE), as the received single-carriersignal is decomposed (spectrally analyzed) into sub-bands, and eachsub-band is separately processed for impairments removal, then thesub-bands are assembled together to form a single carrier, which isfurther processed for phase noise equalization. FIG. 37B shows thedetail of one possible embodiment of the sub-band processors (2×OSSub-band PROC) appearing in the top level SC-MSBE Rx of FIG. 37A. Noticethat although these processors contain (I)FFTs they have nothing to dowith OFDM, as these (I)FFTs are associated with thedecimation/interpolation at the sub-band level of two OS analysis and OSsynthesis filter banks placed back-to-back, according to the principleof operation of the top level SC-MSBE Rx structure of FIG. 37A, asdescribed next.

The top level receiver structure of FIG. 37A and particular sub-bandprocessor version shown in FIG. 37B are not the preferred ones, howeverthese particular versions initiate progressive derivation of our finalpreferred embodiment for the SC-MSBE Rx. The initial embodiment of FIG.37A is based on the cascade of three sub-systems, namely AnalysisFB→sub-bands processing→synthesis FB.

The Analysis FB separates out the channel into sub-bands. Thedemultiplexed sub-bands are processed in the array of MSB processors,having the timing of all sub-bands mutually aligned, and removing otherimpairments, including POL cross-talk. Then the ‘cleaned-up’ sub-bandsare re-assembled to form a cleaned-up overall single-carrier filling theentire channel. This is accomplished by a Synthesis FB, which is thedual of the analysis FB, and may also be efficiently realized using anyof our embodiments of novel 2×OS analysis FB. Each sub-band processor(PROC) unit of the sub-band processor array feeds a corresponding outputof one of the two synthesis FBs for the X and Y POLs, as shown. The i-thsub-band PROC X output feeds the i-th input port of the X-POL synthesisFB, while the i-th sub-band PROC Y output feeds the i-th input port ofthe Y-POL synthesis FB. The output of each synthesis filter bank (oneper POL) is finally processed in a wideband carrier recovery (CR)system, which generates the decisions. Our preferred embodiment for theCR is also MSDD based, but we use a block-parallelized version of thisMSDD called poly-block. One advantage of bringing the signal back tohigher rate (by re-assembling the sub-bands by means of the synthesisfilter bank) prior to applying the MSDD, is that the enhanced samplingrate implies better tolerance to laser phase noise (laser phase noisetolerance is a monotonically decreasing function of the sampling rate,due to the random walk of the phase which picks up more variance overlonger sampling intervals, decorrelating the phase samples and degradingthe quality of the phase estimation).

FIG. 37B presents the detail of the initial sub-band processorembodiment for the SC-MSBE Rx. The first three quarters of thisprocessor are identical to the three quarters of the sub-band Rx used inFIGS. 33,34 for MSB (DFT-S) OFDM. Following estimation and compensationof the IQI, CFO, (NL) PN, and CTO impairments, we perform the 2:1decimation by means of a pair of a 2N-pnt FFT back-to-back with an N-pntFFT, extracting half the oversampled band while dropping the other half,as shown in FIG. 8 (the EXTRACT HIGH/LOW HALF-BAND modules). This isthen followed by adaptive polarization tracking of the time-domainreceived X and Y signals band-passed to a spectral content limited tothe particular 1.66 GHz sub-band. Thus the 2:1 FFT-IFFT based decimatorcompletes the 2×OS filter bank action, cleaning up the imperfection ofthe bandpass filters of the filter-bank. In the remainder of thesub-band processor, having mitigated the polarization and otherimpairments of each sub-band, we proceed to the re-assembly thesub-bands to a complete (cleaned up) single-carrier signal, by means ofthe 2×OS synthesis FB, aided by the preliminary processing alreadyperformed in the back-end of the sub-band processor, namely 1:2 IFFT-FFTbased interpolation (an N-pnt IFFT feeding a 2N-pnt IFFT). Here we alsoneed to perform a cyclic shift of half-band for the odd numberedsub-band indexes, in order to conform with the synthesis FB action shownin FIG. 8. This operation is performed by the ADD into HIGH/LOWHALF-BAND module.

FIG. 38 starts from FIG. 37B and proceeds to derive an alternativeembodiment of the sub-band PROC. Using the linearity of the (I)DFT andthe 2×2 MIMO transformation, we perform the following simplication:

v _(n) ^(X)=DFT₆₄ {W _(XX)·IDFT₆₄ {u _(n) ^(X) }+W _(YX)·IDFT₆₄ {u _(n)^(Y) }}=W _(XX)·DFT₆₄{IDFT₆₄ {u _(n) ^(X) }}+W _(YX)·DFT₆₄{IDFT₆₄ {u_(n) ^(Y) }}=W _(XX) ·u _(n) ^(X) +W _(YX) ·u _(n) ^(Y)

v _(n) ^(Y)=DFT₆₄ {W _(XY)·IDFT₆₄ {u _(n) ^(X) }+W _(YY)·IDFT₆₄ {u _(n)^(Y) }}=W _(XY)·DFT₆₄{IDFT₆₄ {u _(n) ^(X) }}+W _(YY)·DFT₆₄{IDFT₆₄ {u_(n) ^(Y) }}=W _(XY) ·u _(n) ^(X) +W _(YY) ·u _(n) ^(Y)

The significance of this result is that the back to back N-pnt (I)FFTsmay be discarded in FIG. 37B, yielding the system of the current FIG.38, performing the MIMO operation directly on the serialized thehalf-band of the 2N-FFT output, and mapping the two outputs ontohalf-bands of the 2N-pnt IFFT inputs.

FIG. 39 continues the further simplification of the embodiment of thesub-band PROC derived in FIG. 38. Rather than serializing the paralleloutputs of the DROP HIGH/LOW SUB-BAND modules, performing the 2×2 MIMOprocessing at the serialized rate of 1.66 GS/s, and then parallelizingagain for application to the inputs of the ADD into HIGH/LOW HALF-BANDmodule, as in FIG. 38, it is more convenient, as shown in the currentFIG. 39, to parallelize the MIMO operation, applying it onto eachsub-carrier X,Y pair at lower rate by a factor of N (but now require Nsuch parallel 2×2 MIMO modules). The figure then describes the preferredembodiment of the sub-band processor for SC-MSBE, to be used inconjunction with the top level SC-MSBE Rx structure of FIG. 37A. In thefollowing two figures we present an even simpler embodiment of theSC-MSBE Rx top level of FIG. 37A, thus achieving simplification of theoverall SC-MSBE Rx at both at the top level and the sub-processor level.

FIG. 40A presents the top level of our preferred embodiment of theSC-MSBE PDM Rx, applicable for single carrier transmission. FIG. 40Bdetails the preferred embodiment of the associated sub-band processors.This SC-MSBE Rx system is compatible with our Single-Carrier NyquistShaped PDM Tx embodiments of FIG. 35,36, however the SC-MSBE Rx of FIG.40A,B may also compatibly function with any conventional SC PDM Tx whichis DACs based, provided that suitable training sequences are digitallyinserted in the time domain via the DACs to enable the operation of theJoint IQI, CFO, CTO, SFO. As shown in FIG. 40A, with this Rx we achievesingle carrier processing by recombining the individually processedsub-bands into an overall single-carrier broadband signal occupying thewhole channel. However, unlike in the embodiment of the SC-MSBE Rx ofFIG. 37A, the re-assembly of sub-bands is not performed here withsynthesis FBs, but is accomplished using instead a DFT-despreading-likeIFFT (of size K⁻=896 points in this exemplary system), one such IFFT perpolarization. The inputs to this IFFT are derived from the assemblingthe parallel outputs of the X (Y) OFDM sub-band processors. Thus thesynthesis filter banks of FIG. 37A for the two POLs, have been replacedby the K⁻-pnt IFFTs in the current FIG. 40A. The reason this is possibleis that the CTO compensation function in the sub-band processorsre-aligns the spread in timing of the sub-bands, thus the overall delayspread is now well within the CP, therefore the may be used at thispoint to undo the corresponding K⁻-pnt FFT action in the NS-SC Tx ofFIG. 35 or 36.

The sub-bands re-assembly (or synthesis) K⁻-pnt IFFT at the top level ofthe SC-MSBE Rx of FIG. 40A also performs the role of the previously used2N-pnt IFFT in each of the sub-band processors in FIG. 39. Thus, this2N-pnt IFFT in the SC-MSBE sub-band processor embodiment of FIG. 39 maybe simply discarded, yielding the preferred SC-MSBE sub-band processorembodiment of FIG. 40B.

We reiterate that the top level SC-MSBE Rx of FIG. 40A along with theSC-MSBE sub-band processor embodiment of FIG. 40B form our preferredembodiment for single-carrier detection.

Also notice that the mid-sub-band which was interspersed by our NS-SC Txwith the data the inputs into the K₊-pnt FFT is routed to theMid-sub-band processor, hence no sub-band processor is provided for itin the sub-band PROC array, thus the remaining data-carrying sub-bandsbecome contiguous, indicating that the insertion of the pilot sub-bandin the SC Tx of FIG. 35 or 36 does not disturb the invertibility of theK⁻-pnt FFT action in the NS-SC Tx by means of the K⁻-pnt IFFT of thecurrent figure. The outputs of the K⁻-pnt IFFT, carrying in effectsingle carrier transmission, are parallelized and presented to theblock-parallelized CR, just as in FIG. 37A.

Next we present a family of embodiments referred to as bonded-MSBE(B-MSBE). This is a variant of our MSB DFT-S OFDM, enhanced such thatthe number N_(SSC) of DFT-S sub-single-carriers (SSC) need no longercoincide with the number M of sub-bands (in our MSB DFT-S OFDMembodiment each sub-band carried a sub-single-carrier as generated by aprior art DFT-S OFDM Tx with precisely M DFT-S FFTs (same number as thenumber of sub-bands in the OS analysis FB based Rx). The new B-MSBE Rxmay also be viewed as a generalized version of an DFT-S OFDM Rx whereinthe processing of the individual SSCs is carried out by assigningmultiple sub-bands per SSC (rather than precisely one sub-band per SSCas in our earlier MSB DFT-S OFDM embodiment).

FIG. 41 presents an exemplary embodiment of a prior art DFT-S Tx[DFT-Spread OFDM for Fiber Nonlinearity Mitigation Yan Tang, WilliamShieh, and Brian S. Krongold, IEEE PHOTONICS TECHNOLOGY LETTERS, VOL.22, pp. 1250-1252, 2010], enhanced by inclusion of mid-band sparse pilottone injection, used here to generate transmission compatible with forthe novel B-MSBE Rx, presently introduced in FIG. 42 below. Here theDFT-S Tx uses N_(SSC) DFT-S FFTs, i.e., it is configured to generate anumber N_(SSC) (N_(SSC)=2in this exemplary case) of DFT-S sub-singlecarriers which is less than the number M of “MSB sub-bands” used in theB-MSBE Rx of FIG. 42 below. Generally the size of each of the N_(SSC)(here two) DFT-spreading FFTs isK_(SCC)=K⁻/N_(SSC)=(K₊-K₀-K_(ENDS))/N_(SSC)-pnts.

In anticipation of comprehending the reception of this signal by theB-MSBE Rx disclosed in FIG. 42, each DFT-S band generated by the DFT-SOFDM Tx of FIG. 41, carrying a sub-single-carrier (SSC), is to beinterpreted here as effectively having a sub-banded internal structure,i.e. viewed as bonding together M/N_(SSC) sub-bands. The bondedsub-bands form N_(SSC) Sub-Single-Carriers (SSC) (here two of them)which are tightly packed in frequency, as generated by the DFT-spreadingFFTs of the DFT-S Tx of FIG. 41, forming the transmitted optical channel(evidently the prior art DFT-S Tx is not cognizant of the sub-bandstructure, yet its sub-single carriers may effectively be viewed ascomposed of several sub-bands bonded together). However, thistransmission is not to be decoded with a conventional DFT-S Rx, but israther received more conveniently with the novel B-MSBE Rx of the nextfigure.

FIG. 42 presents, according to the teachings of this invention, aBonded-Multi-Sub-Band-Equalized (B-MSBE) Rx compatible with the priorart DFT-S OFDM PDM Tx of FIG. 41, which is viewed as the correspondingB-MSBE Tx, and with the sub-band processors used here for B-MSBE shownin FIG. 40B, coinciding with the SC-MSBE sub-band processors.

The Rx disclosed in this figure differs from the MSB DFT-S OFDM Rx (FIG.32 for the top level and FIG. 34 for the sub-band Rx-s) in that thereception is no longer constrained to require that the number ofprocessed sub-bands coincide with the number of SSCs. Rather, each SSCmay now have a spectral width equal to any integer multiple of thesub-band width, i.e. may contain an integer number of sub-bands. In theexemplary system shown, the number of sub-bands per SSC happens to beM/2, where M is the total number of sub-bands, however other ratios arepossible, e.g. M/4 or M/8 sub-bands per SSC, implying that there are 4or 8 SSCs conveyed per channel. The process of grouping severalsub-bands together to form an SSC is referred to as sub-bands bonding.Remarkably, in this approach we have decoupled the spectral supports ofthe sub-bands and SSCs (which now amount to bonded sub-bands). Thisenables for example extraction of a subset of SSC by a specialized“drop” Rx. Moreover, synthesizing a lower number of SSCs, whilemaintaining the same number (say M=16) of sub-bands per FB, allows todecouple the DSP hardware considerations, e.g., as driven by complexityconsiderations, from the optimization the SSC spectral widths (andnumbers) driven by optimal performance considerations, e.g. improve PAPRand non-linear tolerance. As shown by [DFT-Spread OFDM for FiberNonlinearity Mitigation Yan Tang, William Shieh, and Brian S. Krongold,IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 22, pp. 1250-1252, 2010], thespectral width of the sub-bands in DFT-S OFDM must be carefullyoptimized for best non-linear performance, thus having it constrained tothe precise value S=B/M as dictated by signal processing considerations,as in the MSB DFT-S OFDM embodiment of FIG. 34, might be toorestrictive. The disclosed method enables selecting any number N_(SSC)of sub-single-carriers subject to N_(SSC) dividing the number M ofsub-bands. This provides flexibility and an improved tradeoff betweencomplexity and performance, as the number of sub-bands is typicallyselected according to DSP complexity considerations, whereas the numberof the SSC bands is independently selected according to otherconsiderations such as the PAPR and non-linear tolerance (subject to therequirement that that number of SSC bands which are be a divisor of thenumber M of sub-bands).

We now have a continuum of Rx structures from conventional OFDM viaDFT-S OFDM (viewed as multiplexed transmission of multiple SSCs) all theway to single-carrier transmission, with number of SSCs selected at willto be any divisor of M from 1 to M, and with an underlying sub-bandprocessing structure underneath, which in turn is constructed over anOFDM sub-carrier infrastructure.

FIG. 43A-F presents the Joint IQI, CFO, CTO E&C and its sub-systems andmodules—key elements in the SC-MSBE sub-band PROC.

FIG. 43 presents the JOINT IQI, CFO, CTO E&C and its sub-systems andmodules which are key elements in the MSB (DFT-S) OFDM sub-band Rx andSC-MSBE sub-band PROC of FIGS. 33, 34, 40B. According to the teachingsof this invention, the E&C functionality above should be performed noton per-sub-band basis but rather be jointly performed on pairs ofsub-bands with center frequencies symmetrical around the mid-bandfrequency. Thus, in the bi-polar indexing convention for the sub-bands,whereby the index 0 denotes the sub-band at the mid-band frequency, thetwo sub-bands indexed +i and −i are symmetrically placed aroundzero-frequency of the complex-envelope, and should be processed together(these sub-bands are referred to here as mirror sub-bands). Referring toFIG. 43A, the signals associated with the two minor sub-bands aredenoted r _(k) ^((⊥i)). These signals are degraded IQI, CFO and CTOimpairments and are therefore sequentially passed through the IQI COMP,CFO(+PN) COMP and CTO COMP modules, in this order, for the three typesof impairments to be compensated for. It turns out that the ordering ofthe three compensation modules is critical; the selection of the optimalordering out of the six possible orderings is non-trivial. The reasonwhy impairments compensation order is important is that there iscoupling between the three types of impairments which mutually affectone another, while also affecting the ability to estimate and compensatethese impairments. The procedure for estimation of the operatingparameters to be further input into the three compensator modules(namely estimates of the amounts of IQI, CFO and CTO impairments) occursin the novel module marked ‘JOINT IQI, CFO, CTO Estimation’ and will bedetailed further below. First, let us describe the operation of thethree comp modules. Actually, compensations of CFO and CTO are wellknown (provided good estimates of these parameters have been derived).The CFO is simply compensated by demodulating the signal with anestimate of the time-evolving rotating phasor. This phasor may beavailable from the sparse pilot tone or may be generated from estimatesof the phase increments by a numerically controlled oscillator (NCO),the functionality of which is fully defined by the mapping {circumflexover (θ)}[k′]→exp{jΣ_(k′=0) ^(k){circumflex over (θ)}[k′]}. In this casethe challenge is to estimate the phase increments {circumflex over(θ)}[k′] in the wake of coupling between the CFO target impairment andthe other two impairments, but the estimation task will be addressedbelow. We also mention that the estimated phase associated with CFO iscommon phase equally affecting all sub-bands, thus it is to be equallyapplied for demodulation to both mirror sub-bands, as reflected in theCFO COMP structure which multiplies both outputs of the first IQI COMPblock (corresponding to the two mirror sub-bands) by the conjugate ofthe rotating phasor above. As for CTO compensation, once the amount{circumflex over (τ)}^((±i)) of coarse timing offset has been estimated(in integer units at the sub-band sampling rate) then the CTO COMPmodule simply applies the opposite integer delays −{circumflex over(τ)}^((±i)) as shown. The CFO and CTO estimates are derived from theestimation module D&C based CTO&CFO joint EST’ (D&C stands for“Delay&Correlate”) to be described below.

Describing now the IQI COMP module, this novel module is specified bythe following non-linear transformation of the two sub-band signals,comprising linear combinations but also complex conjugations (CC),denoted by an overbar here (making it non-linear):

R _(k) ^((i))= r _(k) ^((i)) − {circumflex over (γ)}^((−i)) r _(k)^((−i)) =r _(k) ^((i))−{circumflex over (γ)}^((−i)) r _(k) ^((−i))

R _(k) ^((−i))=−{circumflex over (γ)}^((i)) r _(k) ^((i)) +r_(k) ^((−i))  (4)

The parameters {circumflex over (γ)}^((±i)), representing the amounts ofIQI imbalance in the two respective mirror sub-bands, are estimated bythe IQI EST module. The justification for this IQI COMP structure isthat the model for the generation of IQI imbalance impairment in thecoherent optical Rx front-end is as follows. Considering all sub-bandspectral slices as being conceptually superposed to form the opticallydetected broadband signal at the optical Rx FE input, then this signalis written as

$\rho_{k} = {\sum\limits_{i = 1}^{{M/2} - 1}\; {\sum_{\pm}{\rho_{k}^{({\pm i})}.}}}$

In particular, the input sub-bands ρ_(k) ^((=i)) generate their own IQIimpairments at the output of the Rx FE, yielding the following impairedcontributions to the overall broadband output r_(k) of the Rx FE:

ρ_(k) ^((±i)IQI)=α^((±i))ρ_(k) ^((±i))+β^((±i)) ρ_(k) ^((±i)) ∝ρ_(k)^((±i))+γ^((±i)) ρ_(k) ^((±i)) ; γ^((±i))≡β^((±i))/α^((±i))   (5)

The first equality above is a well-known model of IQ imbalance, e.g. [J.Tubbax et al., Proc. GLOBECOM'03,(2003)]. The proportionality relationis obtained by dividing and multiplying by α^((±i)) and introducing thedefinition of γ^((±i)) as a single complex parameter defining theimbalance, as constants α^((±i)) in the E&C chain do not matter (thecarrier recovery and polarization demux modules compensate for anyundermined multiplicative constants in the processing chain). Now thesignal components ρ_(k) ^((±i)IQI) within the broadband signal r_(k) atthe Rx FE output are no longer confined to the respective ±i sub-bands.Indeed, considering first ρ_(k) ^((i)IQI) ∝ρ_(k) ^((i))+γ^((i)) ρ_(k)^((i)) , the CC on ρ_(k) ^((i)) corresponds to conjugate inversion inthe spectral domain, thus the contribution γ^((i)) ρ_(k) ^((i)) willactually appear in the −i^(th) sub-band output after filtering throughthe filter bank. Similarly, inspecting ρ_(k) ^((−i)IQI) ∝ρ_(k)^((−i))+γ^((−i)) ρ_(k) ^((−i)) , the CC on ρ_(k) ^((−i)) implies thatthe contribution γ^((−i)) ρ_(k) ^((−i)) will be spectrally CC-ed andmirrored over and will actually appear in the +i^(th) sub-band outputafter filtering through the filter bank. Thus, the two CC-edcontributions γ^((±i)) ρ_(k) ^((±i)) actually “cross-over” to the minorsub-band output, whereas the contributions ρ_(k) ^((±i)) actually “stayput” appearing in the corresponding sub-bands at the FB output. Itfollows that the following signals are obtained at the two mirrorsub-band outputs of the FB:

r _(k) ^((i))=ρ_(k) ^((i))+γ^((−i)) ρ_(k) ^((−i)) ; r _(k) ^((−i))=ρ_(k)^((−i))+γ^((i)) ρ_(k) ^((i))   (6)

This completes our model, derived for the first time to our knowledge,of the interaction between IQI and sub-band filtering via a FB. At thispoint we may verify that our novel embodiment of IQI COMP as describedin Eq. (4) is indeed able to compensate for the IQI impairment of Eq.(6). This is accomplished by substituting Eqs. (6) into Eqs. (4) andsimplifying, yielding

R _(k) ^((i))=ρ_(k) ^((i))+γ^((−i)) ρ_(k) ^((−i)) −{circumflex over(γ)}^((−i)) [ρ_(k) ^((−i))+γ^((i)) ρ_(k) ^((i)) ]=(1+{circumflex over(γ)}^((−i)) γ^((i)) )ρ_(k) ^((i))+(γ^((−i))−{circumflex over(γ)}^((−i)))ρ_(k) ^((−i))

R _(k) ^((−i))=−{circumflex over (γ)}^((i)) [ρ_(k) ^((i))+γ^((−i)) ρ_(k)^((−i)) ]+ρ_(k) ^((−i))+γ^((i)) ρ_(k) ^((i)) =(γ^((i))−{circumflex over(γ)}^((i)))ρ_(k) ^((i)) +(1+{circumflex over (γ)}^((i)) γ^((−i)) )ρ_(k)^((−i))   (7)

It is now apparent that if our estimates of IQI parameters are precise,i.e. if γ^((±i))≅{circumflex over (γ)}^((±i)) then the coefficients(γ^((±i))−{circumflex over (γ)}^((±i))) multiplying the IQI-cross-talkterms ρ_(k) ^(∓)) in the two equations are then very small, and theamount of IQI-cross-talk will have been suppressed. Ideally, if theestimates are perfect, γ^((±i))={circumflex over (γ)}^((±i)), then weobtain R_(k) ^((±i))∝ρ_(k) ^((±i)), i.e. we attain perfect compensation.

At this point we consider whether the presence of CFO affects the IQICOMP procedure just described. It turns out that it does not, as can beseen from the following argument: CFO due to the offset between theoptical LO frequency and the incoming carrier frequency is a commonphase impairment identically affecting all sub-bands, amounting tomultiplication by the linear phase factor e^(jθk). Thus ρ_(k) ^((±i)) isto be replaced by ρ_(k) ^((±i))e^(jθk) in the derivation above. Definingeffective bandpass signals ρ′_(k) ^((±i))=ρ_(k) ^((±i))e^(jθk) andredoing the derivation above with ρ′_(k) ^((±i)) replacing ρ_(k) ^((±i))everywhere, it is seen that the derivation still holds. The only caveatis whether the ρ′_(k) ^((±i)), which have their spectra shifted fromtheir nominal positions, make it without distortion through the bankfilters. We assume small CFO (as can be achieved by tracking the CFO byfeedback to the laser or digital feedback to a demodulator ahead of theFB, essentially using a PLL), and moreover we recall that the roll-offof the filter banks is not sharp but is mildly sloping out-of-sub-band.Therefore, the slightly shifted spectra, as induced my small CFO, willmake it through the bank filters, and the (I)FFT based decimation willalso allow the slightly shifted spectra through, by virtue of the cyclicprefix guardband. Next, we may consider whether the simultaneouspresence of CTO will affect the IQI COMP operation. In this case we mayagain define new effective signals ρ″_(k) ^((±i))=ρ_(k τ) _((±i))^((±i))e^(jθk) including the different CTO delays on the two sub-bandssignals which propagate at different center frequencies hence experiencedifferent group delays due to CD as well as including the CFO (thedouble-prime indicates the simultaneous presence of CFO and CTO). TheIQI COMP derivation is still valid with ρ_(k) ^((±i)) being replaced bythe effective signals ρ″_(k) ^((±i)).

This completes the proof that the disclosed IQI comp structure indeedachieves compensation of IQI even in the presence of CFO+CTO, providedthat reasonable estimates of the IQI parameters may be obtained. Next wedisclose in FIG. 43B a novel IQI EST module structure, along a noveltraining sequence procedure which are able to generate good estimates ofthe IQI parameters γ^((±i)). In this case it is no longer true that thejoint presence of CFO and CTO will not affect the estimation of IQI(though we have seen that these impairment do not affect thecompensation of ICI).

The novel disclosed IQI EST training sequence procedure actually alsoassists in joint estimation of CFO and CTO, but those aspects will bedescribed further below. To overview the novel data-aided algorithm forjoint estimation of IQI, CTA and CFO, this algorithm is based ontraining sequences with half their spectrum nulled out at a time, firstthe lower half then the upper half, which results in the filter bankdiverting the IQI interference to the mirror-image output port, where itmay be readily estimated by comparison with the unperturbed signal inthe original output port, on which the SCA may be further applied toextract CTO and CFO with high precision. Our joint estimation procedureis based on periodically launching training sequences, transmitted intoeither the upper or lower sub-band while nulling the other sub-band. Atraining sequence {tilde under (A)}_(k) ^((±i)TR) is launched intoeither the +i or −i sub-bands at the Tx but not in both; upon the TSbeing launched in the ±i sub-band then a null is simultaneously launchedin the ∓i minor sub-band. Actually all pairs of signals are treated thisway simultaneously, meaning that TS are first launched into allsub-bands with index i>0 then into all sub-bands with index i<0. As aresult a broadband upper-single-side-band (USSB) occupying half of thechannel spectrum (to the right of the optical carrier) is generated,followed by a broadband lower-single-side-band (LSSB) signal occupyingthe lower half of the channel spectrum (to the left of the opticalcarrier). The USSB launch is followed by the LSSB launch or viceversa.

Each training sequence (TS) launched at the Tx into a particularsub-band port (while nulling out its minor sub-band input) is designedto serve for joint estimation of all three impairments, and inparticular to assist in estimating CTO according to the Delay&Correlate(D&C) principles (e.g. using the Schmidl-Cox Algorithm (SCA) or Minn CTOestimation algorithms used in wireless communication). The trainingsequences then consist of pairs of related consecutive or separatedtraining symbols, in particular for SCA we shall consider for the pairsof identical OFDM symbols or two identical halves of a single OFDMsymbol, with each of the two identical records having length L, whereasfor the Minn algorithm we shall consider four consecutive records withthe first two records identical, and the last two records identical tothe first two up to a sign inversion.

Assuming no IQI is generated at the Tx (and the optical links also donot generate IQI, and also ignoring the link nonlinearity) then wereceive at the Tx input the following pairs of signals, in the spectralranges corresponding +i and −i sub-bands, for the US and LS launchesrespectively: [{tilde under (ρ)}_(k−τ) _((i)) ^((i)), 0]_(USSB),[0,{tilde under (ρ)}_(k−τ) _((i)) ^((−i))]_(LSSB), where thefirst/second component of each pair correspond to the ±i sub-bands.These signals are already affected by CTO as evident in their differentdelays. According to Eq. (6), as a result of the IQI and CFO in the RxFE, the signals received at the two ±i filter bank outputs while therespective TS is on, are given by:

[{tilde under (ρ)}_(k−τ) _((i)) ^((i))e^(jθk), γ^((i)) ρ_(k) ^((i)) e^(−jθk)]_(USSB), [γ^((−i)) ρ_(k) ^((−i)) e^(−jθk), {tilde under(ρ)}_(k−τ) _((−i)) ^((−i))e^(jθk)]_(LSSB)

In contrast, when data is transmitted rather than U/L-SSB sequences, thepair of FB mirror outputs is given, per Eq. (6) by

[ρ_(k) ^((i))+γ^((−i)) ρ_(k) ^((−i)) , ρ_(k) ^((−i))+γ^((i)) ρ_(k)^((i)) ]_(two-sided data)

Significantly, under U/L-SSB launch, the +i/−i sub-band receives itsignal free of IQI, whereas its mirror sub-band −i/+i just receivesinterference from the original sub-band conjugated, scaled by theappropriate IQI coefficient. Thus the conjugate leakage due to the IQimbalance is routed to the mirror-image sub-band, resulting in theoriginal sub-band launched signal and its scaled CC interference beingseparated out to two different sub-band outputs of the FB. In each ofthe two training cases, both filter bank mirror outputs experience asingle delay and a single CFO (albeit of opposite sign). This is incontrast to the case of concurrent excitation of the two filter bankswherein each filter-bank output would contain a mixture of two delaysand a mixture of the two opposite sign CFOs. This suggests simpledata-aided joint processing of the two filter-bank outputs obtainedunder the respective USSB- and LS-launches, jointly estimating IQimbalance, CTO and CFO. We note that the two +i/−i FB outputs are CC ofeach other up to a constant. To estimate the IQ imbalance parameters inthe data-aided training mode, we simply divide one filter bank output bythe CC of the other one, yielding:

$\begin{matrix}\{ \begin{matrix}{{USSB}\text{-}{launch}\text{:}\begin{matrix}{{r_{k}^{({- i})}\text{/}{\overset{\_}{r}}_{\sim k}^{(i)}} = {\lbrack {{\overset{\_}{r}}_{\sim k}^{({- i})}\text{/}r_{\sim k}^{(i)}} \rbrack^{\star} =}} \\{\lbrack {( {e^{j\; \theta \; k}{\overset{\_}{\gamma}}^{(i)}\rho_{\sim {k - \tau^{(i)}}}^{(i)}} )\text{/}( {e^{j\; \theta \; k}\rho_{\sim {k - \tau^{(i)}}}^{(i)}} )} \rbrack^{\star} = \gamma^{(i)}}\end{matrix}} \\{{LSSB}\text{-}{{launch}:\begin{matrix}{{r_{\sim k}^{(i)}\text{/}{\overset{\_}{r}}_{\sim k}^{({- i})}} =} \\{{( {e^{{- j}\; \theta \; k}\gamma^{({- i})}\overset{\_}{\rho_{\sim {k - \tau^{({- i})}}}^{({- i})}}} )\text{/}( {e^{{- j}\; \theta \; k}\overset{\_}{\rho_{\sim {k - \tau^{({- i})}}}^{({- i})}}} )} = \gamma^{({- i})}}\end{matrix}}}\end{matrix}  & (8)\end{matrix}$

This is compactly written as

IQIestim: {circumflex over (γ)}^((±i)) =r _(k) ^((∓i))/{tilde under(r)}_(k) ^((±i))   (9)

Remarkably, the conjugate-divisions cancel out the common delay andcommon CFO present in the two sub-bands under either launch(irrespective of the value of the delay and CFO), while recovering justthe IQ imbalance parameters. As the last equation holds for each of thediscrete-time instants, k, over the duration of the US/LS-launchtraining sequence, and since the noises in the different samples arelargely independent, it is possible to average the conjugate ratios overthe 2 L samples of the training sequence duration, yielding thefollowing improved estimates for the IQ imbalance parameters in the twosub-bands:

$\begin{matrix}{{{{{USSB}\text{-}{launch}\text{:}\mspace{14mu} {\hat{\gamma}}^{(i)}} = {\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {r_{k}^{({- i})}/{\overset{\_}{r}}_{\sim r}^{(i)}}}}};}{{{LSSB}\text{-}{launch}\text{:}\mspace{14mu} {\hat{\gamma}}^{({- i})}} \equiv {\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {r_{k}^{(i)}/{\overset{\_}{r}}_{\sim r}^{({- i})}}}}}} & (10)\end{matrix}$

where T is the total duration (in discrete-time units) of the trainingsequence (e.g., consisting of two symbols for the SCA (T=2 L) and offour symbols for the Minn Algorithm (T=4 L)).

This completes the mathematical description of the IQI EST module, theblock diagram of which is shown in FIG. 43B. In that sub-figure thedivider blocks are defined as taking the ratio of the left input toright input, and the MA blocks are moving averages. Notice that theestimates are valid while the proper training sequences are on, asindicated.

It remains to describe the joint estimation of CFO and CTO, occurring inthe ‘D&C based CTO&CFO joint EST’ module, for which two possiblealternative embodiments are detailed in FIGS. 43C,D. In turn these novelmodules resort to the ‘Autocorr. based metrics’ prior art modulesdescribed in FIGS. 43E,F for the two alternative algorithms ofSchmidl-Cox algorithm (SCA) and Minn algorithm. The SCA module of FIG.43E generates the complex-valued moving average autocorrelation (MA-ACC)Γ_({tilde under (r)}) _(k) [k; L; L]=Σ_(m=0) ^(L−1){tilde under(r)}_(k−m){tilde under (r)}*_(k−m−L) as well as the real-valued movingaverage power (MA-PWR)

${\Gamma_{r_{\sim k}}\lbrack {k;0;L} \rbrack} = {\sum\limits_{m = 0}^{L - 1}\; {r_{\sim {k - m}}}^{2}}$

where these formulas are particular forms of the following generalmoving average autocorrelation definition with a lag of L over a movingwindow W:

${\Gamma_{r_{\sim k}}\lbrack {k;L;W} \rbrack} \equiv {\sum\limits_{m = 0}^{W - 1}\; {r_{\sim {k - m}}{r_{\sim {k - L - m}}^{*}.}}}$

For the Minn algorithm module of FIG. 43F, the two output metrics, alsoreferred to as MA-ACC and MA-PWR are generalized to mean the movingaverage autocorrelation and power metrics as generated under the SCAfurther filtered through the L-pnt delay&add impulse response δ[k]+δ[k−2L] (in effect this filtering amounts to adding the SCAautocorrelation/power to a 2 L-pnt delay version of itself).

Back to FIGS. 43C,D notice that initially each of the twoIQI-compensated inputs R_(k) ^((±i)) are individually applied to its own‘Autocorr. based metrics’ block. The normalized absolute-squared MA-ACCis generated by calculating |MA-ACC|²/|MA-PWR|². The two embodiments 43Cand 43D differ in the order in which the division and absolute squaringare performed, but yield identical results for the normalizedabsolute-squared MA-ACC, which is input into the argmax operation,determining the time-index k whereat this normalized metric peaks to itshighest value. The output of the argmax operation provides the sought{circumflex over (τ)}^((±i)) CTO estimate. As for the estimation ofphase, this is also specified in the SCA and Minn algorithm in terms ofthe angle (complex number argument) of the complex-valued MA-ACC (itdoes not matter if the angle is extracted prior to or afternormalization). The extracted angle is to be divided by L, the laginvolved in repeating the training symbols, in order to extract theangle estimate {circumflex over (θ)}. Our embodiments of FIG. 43C,D aredistinguished from plain processing of the MA-ACC and MA-PWR metricsunder the prior art D&R joint CTO and CFO estimation, in that ourembodiments also perform joint processing of the CFO estimates derivedfrom the two sub-band signals by averaging over the two sub-bands (whichis an aspect unique to our FB approach). To this end there are twoalternatives, as detailed in 43C,D: The first alternative shown in C isto average across the two minor sub-bands either in the real-valuedangular domain, extracting the angle of MA-ACC first, then taking thearithmetic average of the two angles (the ½ of arithmetic average isabsorbed with the 1/L normalization). The second alternative shown in Dis to average the MA-ACCs in the complex domain (actually just sum themup) then extract the angle. The performance of these two alternativeangle averaging scheme is very similar and results in roughly 3 dB ofSNR improvement vs. not averaging over the two mirror sub-bands.

We now consider the usage of the feedforward phase estimate, ψ₀ ^(P)[k],input at the bottom of the JOINT IQI, CFO, CTO E&C module in order todemodulate CFO and (NL) phase noise. As shown in FIG. 43A this estimateis exponentiated, generating e^(jψ) ⁰ ^(P) ^([k]), which complex signalis input into the CFO(+PN) COMP module in order to demodulate the {tildeunder (R)}_(k) ^((±i)) sub-band signals (which already underwent IQIcompensation). This demodulating signal is alternative to that obtainedfrom the D&C based CTO&CFO joint EST signal, which is accumulated andexponentiated. The optional use of one the two signals is shown by meansof the switch feeding the exponentiation module. Actually, it ispossible to mix the two signals, switching

between them or even generate weighted moving average weightedcombinations, though the simplest strategy is to use one signal or theother. The feedforward common phase signal ψ₀ ^(P)[k] is available rightaway, and can also correct other types of phase impairments in additionto CFO, as explained in FIG. 45, thus in the preferred embodiment weshall use this signal for demodulation, using the D&C based algorithmsolely for CTO estimation (meaning that the portions of FIG. 43C,D usingthe angle extractors (∠) may be removed.

This completes the description of all modules in FIG. 43A-43F.

Synchronization from IQI EST independent of the CTO EST

One issue which arises is how the receiver knows when the two types oftraining sequences (U/L-SSB) are transmitted, as only then does the IQIEST module generate a valid output one of its output ports or another.One could suggest that the output of the CTO module provides propersynchronization of the training sequences, however this seems somewhatof a ‘chicken&egg’ problem, as unless IQI is compensated first then theresults of the CTO estimation are not accurate, and viceversa. However,the system may still converge as even with an imperfect CTO estimate themoving average of Eq. (10) will yield a relatively precise estimate ofIQI even if the averaging window is somewhat off (provided the window issufficiently long). Once this estimate is derived, the next TS launchwill experience less IQI and therefore the CTO estimate will improve,and so forth, indicating potential convergence of the system despite thecoupling between the two impairment types.

Yet another approach to synchronizing the TS window for IQI estimationis to realize that the event of SSB TS transmission may be detected bythe FB based system which acts as a coarse spectrum analyzer. Thus, astrong imbalance in the powers detected in minor sub-bands is anindication of the presence of training sequence. In fact we alreadyprovide the computational means to detect such imbalance, as we take theconjugate ratio of the signals in mirror sub-bands and average allsignals. If we were to monitor the powers (absolute squares) of theaveraged ratios corresponding to our estimates {circumflex over(γ)}^((±i)), i.e. the quantities |{circumflex over (γ)}^((±i))|² or evenbetter their sum, then we would experience a sharp drop in valuewhenever the moving average window coincides with the TS window. Tounderstand this notice that from Eq. (6) it follows that the ratio r_(k)^((−i))/{tilde under (r)}_(k) ^((i)) (or its minor image obtained bysubstituting −i for I and conjugating) is generally expressed as

${{\hat{\gamma}}_{k}^{(i)} \equiv {r_{k}^{({- i})}/{\overset{\_}{r}}_{\sim k}^{(i)}}} = {\frac{\rho_{k}^{({- i})} + {\gamma^{(i)}\overset{\_}{\rho_{k}^{(i)}}}}{\overset{\_}{\rho_{k}^{(i)}} + {\overset{\_}{\gamma^{({- i})}}\rho_{k}^{({- i})}}}.}$

(Here we do not interpret {circumflex over (γ)}_(k) ^((i)) as IQimbalance but rather as the metric obtained by taking the indicatedratio).

When SSB signals are generated, this expression reduces to Eqs. (8) asseen above. However, when the data is on and the launch is two sided,then this expression is approximated by dropping the IQI interferenceterms (as the γ^((=i)) coefficients are small).

$\begin{matrix}{{{{\hat{\gamma}}_{k}^{(i)}}^{2} \equiv {\frac{1}{T}{{\sum\limits_{k = 0}^{T - 1}\; {r_{k}^{({- i})}/{\overset{\_}{r}}_{\sim k}^{(i)}}}}^{2}}} = {{\frac{1}{T}{{\sum\limits_{k = 0}^{T - 1}\; \frac{\rho_{k}^{({- i})} + {\gamma^{(i)}\overset{\_}{\rho_{k}^{(i)}}}}{\overset{\_}{\rho_{k}^{(i)}} + {\overset{\_}{\gamma^{({- i})}}\rho_{k}^{({- i})}}}}}^{2}} \cong {\frac{1}{T}{{\sum\limits_{k = 0}^{T - 1}\; \frac{\rho_{k}^{({- i})}}{\overset{\_}{\rho_{k}^{(i)}}}}}^{2}}}} & (11)\end{matrix}$

The power of this squared ratio signal, still fluctuates, mainly underthe influence of the randomness of the transmitted signal, however theaverage value of this ratio is unity, and under the moving average, thissignal is likely to converge to unity with fairly high accuracy.

Alternatively, we may generate the absolute square of thesample-by-sample ratio, then apply an MA, in effect reversing the orderof squaring and averaging:

$\begin{matrix}{{G_{k}^{(i)} \equiv {\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {{r_{k}^{({- i})}/{\overset{\_}{r}}_{\sim k}^{(i)}}}^{2}}}} = {{\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {\frac{\rho_{k}^{({- i})} + {\gamma^{(i)}\overset{\_}{\rho_{k}^{(i)}}}}{\overset{\_}{\rho_{k}^{(i)}} + {\overset{\_}{\gamma^{({- i})}}\rho_{k}^{({- i})}}}}^{2}}} \cong {\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {\frac{\rho_{k}^{({- i})}}{\overset{\_}{\rho_{k}^{(i)}}}}^{2}}}}} & (12)\end{matrix}$

Thus, when we are perfectly synchronized with the TS then either ofthese signals will be low whereas when when there is no overlap with theDS then either of these signals will be very close to unity. Thus, asthe moving window advances, the system will go through a minimum, whichwill indicate when the window is synched, then it will come back upagain/

By generating the mirror image ratios,

${{\hat{\gamma}}_{k}^{({- i})}}^{2} \equiv {\frac{1}{T}{{\sum\limits_{k = 0}^{T - 1}\; {r_{k}^{(i)}/{\overset{\_}{r}}_{\sim k}^{({- i})}}}}^{2}\mspace{14mu} {or}\mspace{14mu} G_{k}^{({- i})}} \equiv {\frac{1}{T}{\sum\limits_{k = 0}^{T - 1}\; {{r_{k}^{(i)}/{\overset{\_}{r}}_{\sim k}^{({- i})}}}^{2}}}$

we may obtain the timing under the LSSB launch, which corresponds to theCTO for the −I sub-band. Thus, we have two alternative CTO estimates:

|{circumflex over (τ)}_(γ) ^((±i))|²≡arg_(k)min{|{circumflex over(γ)}_(k) ^((±i))|²}; |{circumflex over (τ)}_(G)^((±i))|²≡arg_(k)min{|G_(k) ^((±i))|²}  (13)

The description above discloses a novel method to derive CTO by analternative method differing from the D&C based methods (SCA and Minnalgorithms), free of the chicken&egg coupling between CTO and IQI.

In additional embodiments (not shown) it is possible to further mergethis CTO estimation method with the D&C methods (SCA and Minn algorithm)e.g. by subtracting the moving average metrics of Eq. (11) and (12) fromthe D&C based metrics and taking the argmax of the difference in orderto derive improved coarse timing estimates.

FIG. 44A-44B presents a a block diagram extending the IQI of FIG. 43B toprovide joint IQI+CTO by implementing the CTO procedure of Eqs. (13)along with Eqs. (11) (12). The block diagrams of FIG. 44A,B preciselyimplements the functions described in these equations. In FIG. 44C weembed the IQI+CTO EST module embodiments of FIG. 44A,B into the completeJOINT IQI, CFO, CTO E&C system, obtaining a second embodiment of thissystem, differing from the one we disclosed in FIG. 43A. Notice that theCFO+PN estimation is derived in the system of FIG. 44C from thefeedforward common phase estimate as generated in the next FIG. 45.

FIG. 45 shows the detailed internal block diagram of the mid-sub-bandPROC appearing in the top level views of the MSB (DFT-S) OFDM andSC-MSBE Rx shown in FIGS. 37, 40A and 42A. This figure also details allthe interfaces of the mid-sub-band PROC to other modules in the Rxsystem. The mid-sub-band PROC is fed by the 0-th sub-band output signalsfrom the X-POL and Y-POL OS analysis filter banks. These sub-bands carrythe received signal due to sparse pilot signals launched in thecorresponding mid-sub-band of each polarization at the Tx.

Detailing the Tx-side sparse pilot generation, the mid-sub-band (in oursub-bands bi-polar labeling convention the 0^(th) sub-band) for each ofthe X-POL and Y-POL paths at the Tx is fed by a pilot signal positionedin frequency either at the center (DC) or somewhat offset from thecenter of the sub-band. The rest of the band is null. This is achievedfor OFDM transmission (FIG. 31) by setting one of the OFDM sub-carriers,at or close to the center of the the mid-sub-band to a non-zeroamplitude (say a real-valued one) A_(p), while the other sub-carriers inthe mid-sub-band are set to null. For DFT-S OFDM transmission (FIG. 33),the sparse pilot tone launch consists of removing the N-pnt DFT-S FFTahead of the MN-pnt IFFT for the particular mid-sub-band, and alsosetting one of the OFDM sub-carriers, at or close to the center of thethe mid-sub-band (center segment of the IFFT inputs) to a non-zeroamplitude (say a real-valued one), while the other sub-carriers in themid-sub-band are set to null. This describes the case where the sparsepilot tone (including its sparse guardband) fills up the entiremid-sub-band. It is also possible for the sparse pilot tone to fill upjust a portion of the mid-sub-band, with the rest of the sub-band filledup with data-carrying sub-carriers. This case, referred to as ‘partialsparse pilot’, allows a variable tradeoff between the quality of (NL)phase noise mitigation and the system spectral efficiency.

In order to mitigate potential generation of Stimulated BrilloinScattering (SBS) due to the intense pilot tone, we may optionally applyat the Tx dithering of the laser frequency, typically at a rate of atleast several MHz. This would yield the functional form e^(j(θ) ^(p)^(k+β sin(θ) ^(d) ^(k))) for the transmitted sparse pilot (momentarilyassuming an ideal laser source). Another mechanism to generate a similarphase dithering may also result not from intentional SBS mitigation, butrather due to the internal operation of the coherent tunable lasersource used as the laser source (tunable laser sources are often lockedto an etalon the free spectral range of which is dithered in order toenable locking the laser frequency). A similar phase dither term may bepresent at the Rx, due to the LO laser etalon locking. Thus, overall wemay end up at the Rx with several phase dither terms superposed,corresponding to the Tx and Rx laser etalons and the SBS suppression

Σ_(l) β_(d) ^(Tx)[l] sin(θ_(d) ^(Tx)[l]k+φ_(d) ^(Tx)[l])

Considering now the structure for reception and processing of the pilottone in the Rx, main processing path in the mid-sub-band PROC receivesthe mid-sub-band signal r₀ ^(X/Y)[k] of each POL (we follow the Y-POL inthe drawing, as for the X-POL the processing structure is identical) andapplies IQI COMP, correcting for the IQ imbalance of this sub-band. Forsimplicity the preferred embodiment of the mid-band IQI COMP does notgenerate its own estimation of the IQI parameter {circumflex over(γ)}⁽⁰⁾. Rather this parameter is simply obtained as the arithmeticmean) {circumflex over (γ)}⁽⁰⁾=1/2({circumflex over (γ)}⁽¹⁾+{circumflexover (γ)}⁽⁻¹⁾) of the two IQI parameter estimates from the two i=±1neighboring sub-bands. In the optional case of partial sparse pilot(i.e., when the sparse pilot fills up just part of the mid-sub-band) alow-pass or band-pass filter is further inserted to extract perturbedpilot band (this is the pilot and the sparse band around it, as injectedat the Tx, perturbed over propagation by noise and impairments)generating a signal denoted by ρ₀ ^(X/Y)[k]. The phase ∠ρ₀ ^(X/Y)[k] ofthis signal is extracted and passed to the ‘pilot offset+optional ditherphase cancellation stage’ which generates and subtracts phase waveformcorrections related to the pilot frequency offset (the pilot subcarrierspectral position relative to the sub-carrier at DC which is mapped tothe Tx laser carrier) and optional dither phase if present, as specifiedfurther below. The resulting phase ψ₀ ^(P)[k] is output as a feedforwardcommon phase estimate to all M−2 sub-band Rx-s/PROCs providing (NL)phase & CFO compensation per sub-band.

In addition, the common phase signal is also applied to the stage namedCFO PLLs which provides a pair of parallel phase-locked-loops (PLL)estimating CFO by heavily low-pass filtering the common phase signal ψ₀^(P)[k] (which includes the phase ramps due to the CFO terms) feedingback the filtered CFO estimates to the LO laser analog frequency inputand to a digital demodulator at the input of the filter bank.

Notice that this method is a feedforward one, requiring precisesynchronization of the paths experienced by the i-th sub-band and thezeroth sub-band such that they arrive aligned in time, allowing tocancel the common phases. By applying sufficient appropriate relativedelays to the data sub-bands and the mid-sub-band output ports of theanalysis FB, this feedforward time-alignment condition may be readilymet for all the sub-bands.

We now analyze the phase noise and frequency offset mechanisms affectingthe phase of the perturbed pilot, explaining why this mid-sub-bandprocessor, operating in conjunction with the processing activated by itin the rest of the system, substantially mitigates phase noise and CFO.In order to improve SNR, it is desirable to have the amplitude A_(p) ofthe sparse pilot tone as high as possible, at least higher than theaverage amplitude of the data-carrying sub-carriers.

As a result of optical propagation, the sparse pilot acquires opticalphase noise and amplitude modulation, due to the transmit and receive(LO) laser phase noises as well as due to the linear and non-linearphase noises induced in the optical link, the received sparse pilotsignal is given by ρ₀ ^(X/Y)[k]=A₀[k]e^(jψ) ⁰ ^(X/Y) ^([k]), where A₀[k]is amplitude noise and ψ₀ ^(X/Y)[k] is the total phase noise experiencedby the perturbed pilot band, expressed as the sum of multiplecontributions as follows (discarding the superscript X/Y which ought tohave labeled this phase):

ψ₀[k]≡θ_(CFO)k+θ_(p)k+Σ_(l)β_(d) ^(Tx)[l] sin(θ_(d) ^(Tx)[l]k+φ_(d)^(Tx)[l])+φ₀ ^(X/Y-FWM)[k]+φ₀ ^(X/Y-SPM)[k]+φ₀ ^(X/Y-XPM)[k]+φ₀^(X/Y-ASE)[k]+φ₀ ^(Tx-LPN)[k]+φ₀ ^(Rx-LPN)[k]  (14)

where θ_(p)k is the phase ramp due to the frequency offset of the pilottone frequency relative to the Tx laser frequency; θ_(CFO)k is the phaseramp due to the offset of the Tx laser frequency relative to the LOlaser frequency (thus the frequency offset between the received pilotand the LO induces a phase increment of θ_(CFO)+θ_(p) per discrete-timeunit); β_(d) ^(Tx) sin(θ_(d) ^(Tx)k) and β_(d) ^(Rx) sin(θ_(d) ^(Rx) k)are the phase dithers applied at the Tx (either for SBS suppression ordue to etalon dithering in the Tx laser and at the Rx due to etalondithering in the LO laser); φ₀ ^(X/Y-FWM)[k], φ₀ ^(X/Y-SPM)[k], φ₀^(X/Y-XPM)[k] are respectively non-linear phase noises (NL-PN) due tointer-sub-bands Four-Wave-Mixing (FWM), intra-mid-sub-bandSelf-Phase-Modulation (SPM) and inter-sub-bands Cross-Phase Modulation(XPM) all induced within the zero-th sub-band to the interaction of allother sub-bands of the given channel as well as all co-propagatingchannels in the WDM multiplex; φ₀ ^(X/Y-ASE)[k] is linear phase noiseinduced due to the ASE additive noise injected by all optical amplifiersalong the link; φ₀ ^(Tx-LPN)[k], φ₀ ^(Rx-LPN)[k] are the laser phasenoise induced by the Tx laser and the LO Rx laser. Moreover, the termθ_(p)k , associated with the offset of the mid-band pilot tone isevidently missing in this expression.

If the total phase impairment were common to all the sub-bands, then wewould have ideal phase noise and frequency offset cancellation. However,in practice, the phase noises affecting the various sub-bands are notfully correlated. Let us represent the phase and frequency offsetimpairment of the i-th sub-band by conducting a similar analysis for thetotal phase noise ψ_(i) ^(X/Y)[k] affecting the i-th sub-band signalρ_(i) ^(X/Y)[k]=A_(i)[k]e^(jψ) ^(i) ^(X/Y) ^([k]) which is output by theFB, which is expressed as follows (discarding the superscript X/Y whichought to have labeled this phase):

ψ_(i)[k]≡θ_(CFO)k+Σ_(l)β_(d) ^(Tx)[l] sin(θ_(d) ^(Tx)[l]k+φ_(d)^(Tx)[l])+φ_(i) ^(X/Y-FWM)[k]+φ_(i) ^(X/Y-SPM)[k]+φ₀ ^(X/Y-XPM)[k]+φ_(i)^(X/Y-ASE)[k]+φ₀ ^(Tx-LPN)[k−τ^((i))]+φ₀ ^(Rx-LPN)[k]  (15)

Notice here that some of the subscripts, but not all, have been modifiedfrom 0 to i, indicative of some the noise sources being sub-banddependent; those phase noise sources which are constant across allsub-bands qualify as ‘common phase’ (in the sub-band domain). Inparticular, the LO laser phase noise φ₀ ^(Rx-LPN)[k] is a common phasecontribution as it equally affects all sub-bands by the nature of themodulation process, as the time-domain superposition of receivedsub-bands is mixed with the LO laser field ∝ exp{jφ₀ ^(Rx-LPN)[k]},meaning each sub-band is individually mixed by the LO laser field.However, the Tx laser phase noise contributions are not identical in allreceived sub-bands, but are relatively delayed across sub-bands, asindicated by the argument of the Tx-LPN expression φ₀^(Tx-LPN)[k−τ^((i))]. Indeed, while Tx-LPN is common to all transmittedsub-bands (by the argument made above for the LO), nevertheless, in thecourse of optical propagation, due to CD, each of the sub-bandsaccumulates a different delay, τ^((i)) for the i-th sub-band. Thus,simultaneous arrivals at the Rx in different sub-bands at discrete-timek correspond to relatively delayed temporal samples of the Tx laserphase noise. It follows that there is dependence of the Tx-LPNcontributions at the Rx on the index i via the delays τ^((i)). Alsonotice that the CFO and pilot frequency offsets and the Tx and Rx phasedithers are all common phase contributions. Finally, we may verify thatby setting i=0 in Eq. (15) we retrieve Eq. (14) (14)

, as we have implicitly assumed without loss of generality that τ⁽⁰⁾=0,as the zeroth sub-band delay is taken as the time origin at the Rx.

Another key term which to a very good approximation constitutes commonphase equally affecting all the sub-bands is the inter-sub-band XPMwhich depends on the total power of all ‘the other’ sub-bands (i.e. forthe effect on the i-th sub-band, the powers of all the sub-bands with anindex different than i should be summed up. The total power affectingsub-band i may be expressed as the total power of the full WDM signalminus the power of the i-th sub-band. As the WDM signal includesmultiple channels each having multiple sub-bands, the power of a singlesub-band may be of the order of 1% of the total power of the WDM signal,thus the total power affecting each of the sub-bands is nearly constant.This justifies why we were able to set the index of the XPM phase noiseterm to 0.

Finally notice that it is just the non-linear phase noise contributionswhich have a dependence on polarization.

We now describe the functionality and operation of the ‘pilotoffset+optional dither phase cancellation stage’. The role of thismodule, operating in the angular domain, after the ‘angle extract’operation, is to cancel out the phase term θ_(p)k out of ψ₀[k] (Eq.(14)) yielding the reduced pilot phase output, labeled by a P subscript:

ψ₀ ^(P) [k]=ψ ₀ [k]−θ _(p) k=θ _(CFO) k+Σ _(l)β_(d) ^(Tx) [l] sin(θ_(d)^(Tx) [l]k+φ _(d) ^(Tx) [l])+φ₀ ^(X/Y-FWM) [k]+φ ₀ ^(X/Y-SPM) [k]+φ ₀^(X/Y-XPM) [k]+φ ₀ ^(X/Y-ASE) [k]+φ ₀ ^(Tx-LPN) [k]+φ ₀ ^(Rx-LPN)[k]  (16)

The phase ramp term θ_(p)k associated with the carrier frequency offset,which was canceled out is deterministically known, as the phase offsetθ_(p) is determined by the pilot sub-carrier index, counted relative tothe DC sub-carrier, as follows:

θ_(p)=(sparse pilot sub-carrier index)·2π/(IFFT size)

As for the sinusoidal phase modulation dither terms, those also haveknown frequencies as the dither is deterministically induced (butunknown amplitudes and phases due to unknown propagation losses anddelays) therefore a pair of narrowband offset PLLs may readily lock ontothese phases and derive accurate estimates of these phase domainsinusoidal waveforms. Therefore the entire signal θ_(p)k+Σ_(l)β_(d)^(Tx)[l] sin(θ_(d) ^(Tx)[l]k+φ_(d) ^(Tx)[l]) is readily estimated and issubtracted out as indicated in Eq. (16), yielding the reduced commonphase estimate φ₀ ^(P)[k] to be passed on to each of the sub-bands fordemodulation, and also to be input into the CFO PLLs stage of themid-sub-band PROC. The processing in the i-th sub-band has already beendescribed in FIG. 43A which is revisited in order to derive the residualphase noise after CFO(+PN) COMP, in light of the analysis above.

The processed sub-band signals {tilde under (R)}_(k) ^((±i)) are bothmultiplied by the conjugate of the exponentiated feedforward commonphase estimate e^(jφ) ⁰ ^(P) ^([k]), yielding at the CFO(+PN) COMP inFIG. 44A, (following just the positive index, +i):

{tilde under (R)} _(k) ^((i)) e ^(−jφ) ⁰ ^(P) ^([k]) =|{tilde under (R)}_(k) ^((i)) |e ^(jψ) ^(i) ^([k]) e ^(−jψ) ^(i) ^([k]) =|{tilde under(R)} _(k) ^((i)) |e ^(jΔψ) ^(i) ^([k]); Δψ_(i) [k]≡ψ _(i) [k]−ψ ₀ ^(P)[k]  (17)

The demodulation then amounts to subtracting the reduced phase of the0^(th) sub-band as generated by the mid-sub-band PROC (Eq. (16)) out ofthe total phase of the i-th sub-band (Eq. (15)). Performing thesubtraction of the two equations term by term, it is apparent thatseveral terms cancel out, yielding the following residual phasedifference in the i-th output of the CFO(+PN) COMP stage:

Δψ_(i) [k]≡ψ _(i) [i]−ψ ₀ ^(P) [k]=Δφ _(i) ^(X/Y-FWM) [k]+Δφ _(i)^(X/Y-SPM) [k]+Δφ _(i) ^(X/Y-ASE) [k]+Δφ _(i) ^(Tx-LPN) [k]  (18)

where Δφ_(i) ^(X/Y-FWM)[k]≡φ_(i) ^(X/Y-FWM)[k]−φ₀ ^(X/Y-FWM)[k];

Δφ_(i) ^(X/Y-SPM)[k]≡φ_(i) ^(X/Y-SPM)[k]−φ₀ ^(X/Y-SPM)[k];

Δφ_(i) ^(X/Y-ASE)[k]≡φ_(i) ^(X/Y-ASE)[k]−φ₀ ^(X/Y-ASE)[k];

Δφ_(i) ^(Tx-LPN)[k]≡φ_(i) ^(Tx-LPN)[k−τ^((i))]−φ₀ ^(Tx-LPN)[k]

The terms which have been cancelled out are the phases common to allsub-bands, namely the CFO, the phase dithers, the XPM phase noise andthe Rx LO laser phase noise. Significantly, the cancellation of the RxLO LPN implies that LO equalization enhanced phase noise (LO-EEPN) isgone. LO-EEPN is a type of phase noise impairment in conventionalcoherent receivers, whereby the CD group delay equalization in the Rxgenerates extra phase noise due to the LO LPN being delayed differentlyat each frequency in the process of CD equalization. In principle,LO-EEPN is also present in the sub-band processing based Rx-s, wheneverthere is LO PN, as the CTO varies per sub-band, hence the CTO compinduces different delays onto the Rx LO LPN (notice that the Tx LPNexperiences no net enhancement due to CD equalization, since the varioustransmitted sub-bands are relatively delayed in the fiber link but theyare then relatively delayed in the opposite sense in the CTO COMPmodules of each sub-band, such that the net delay on the Tx phase noiseis zero). Now, by virtue of using the sparse pilot tone, the Rx LO LPNis nearly ideally cancelled hence the EEPN impairment is essentiallyabsent from our Rx-s.

The four phase noise terms which are not cancelled out, and may be evenenhanced, are the FWM, the SPM, the ASE and the Tx-LPN. As the two FWMterms in sub-bands 0 and i are correlated, their substraction Δφ_(i)^(X/Y-FWM)[k] may result in either smaller or larger variance of thefluctuations, depending on their coefficient of correlation. AdditionalFWM cancellation measures, outside the scope of this application may beapplied. As for the SPM terms, these are uncorrelated, as each isgenerated solely from subcarriers in its own band, hence the fluctuationvariance of Δφ_(i) ^(X/Y-SPM)[k] is doubled due to the pilot phasesubtraction, however the SPM contribution may be small to begin with,hence this term might not significant.

As for the two ASE terms subtracted to yield Δφ_(i) ^(X/Y-ASE)[k], theseare also uncorrelated, however by selecting the power of the sub-carrierto be at least as large as the average power of each data sub-band, thisterm may be made quite negligible. E.g., if the sparse pilot power istaken to be precisely equal to the average power of each data sub-band,then that implies that the sparse pilot amplitude be √{square root over(N)} larger than the RMS amplitude of the data sub-carriers, where N isthe number of sub-carriers per sub-band, e.g. N=64. Therefore, for OFDMdetection, the linear phase noise variance due to the additive circularGaussian noise added to the sub-carrier phasors, is N times smaller forthe sparse pilot than for the i-th data sub-band, and the ASE-inducedLPN enhancement factor is just 1+N⁻¹, which is negligible for a largevalue of N.

The remaining Tx-LPN term Δφ_(i) ^(Tx-LPN)[k]=φ_(i)^(Tx-LPN)[k−τ^((i))]−φ₀ ^(Tx-LPN)[k] has an interesting behavior due tothe various delays. To see this we substitute Eq. (18) into Eq. (17),yielding the following input into the CTO COMP module,

|{tilde under (R)} _(k) ^((i)) |e ^(jΔψ) ^(i) ^([k]) =|{tilde under (R)}_(k) ^((i)) |e ^(jΔφ) ^(i) ^(X/Y-FWM) ^([k]) e ^(jΔφ) ^(i) ^(X/Y-SPM)^([k]) , e ^(jΔφ) ^(i) ^(X/Y-ASE) ^([k]) , e ^(jΔφ) ^(i) ^(Tx-LPN)^([k])

therefore the CTO COMP output is a time-advance of this signal(substituting k+τ^((i)) for k) as follows

|{tilde under (R)} _(k+τ) _((i)) ^((i)) |e ^(jΔψ) ^(i) ^([k+τ) ^((i))^(]) =|{tilde under (R)} _(k+τ) _((i)) ^((i)) |e ^(jΔφ) ^(i) ^(X/Y-FWM)^([k−τ) ^((i)) ^(]) e ^(jΔφ) ^(i) ^(X/Y-SPM) ^([k+τ) ^((i)) ^(]) , e^(jΔ) ^(i) ^(X/Y-ASE) ^([k+τ) ^((i)) ^(]) , e ^(jΔφ) ^(i) ^(Tx-LPN)^([k+τ) ^((i)) ^(])

The time-advance has no significant impact on the four terms on the RHSof the last equation but it does wield an interesting effect onto thelast Tx-LPN term, which is singled out below:

e ^(jΔφ) ^(i) ^(Tx-LPN) ^([k+τ) ^((i)) ^(]) =e ^(j{φ) ^(i) ^(Tx-LPN)^([k−τ) ^((i)) ^(]−φ) ⁰ ^(Tx-LPN) ^([k]})|_(k→k+τ) _((i)) =e ^(j{φ) ^(i)^(Tx-LPN) ^([k]−φ) ⁰ ^(Tx-LPN) ^([k+τ) ^((i)) ^(]})  (19)

It is now seen that the Tx-LPN of the i-th sub-band has beentime-realigned but now the subtracted Tx-LPN of the mid-sub-band, φ₀^(Tx-LPN)[k+τ^((i))] has been advanced (negative delayed) in time. Theoverall phase noise is now the sum of these two contributions which tendto be more uncorrelated the longer the delay τ^((i)) is, i.e., thehigher the absolute value |i| is. Thus more remoted from the centerfrequency sub-bands tend to generated more phase noise.

In fact, this phase noise impairment is the dual of the LO-EEPN phasenoise impairment described above, and may be referred to as Tx-EEPNphase noise impairment. So, we see that while we have suppressed LO PNaltogether, we have replaced the LO-EEPN impairment in our system by ananalogous Tx-EEPN impairment. Now the LO phase noise has been eliminatedaltogether, but in a sense the zeroth sub-band plays the role previouslyplayed by the LO, as the zeroth sub-band is used to demodulate thereceived signal, which contains Tx phase noise (but the zeroth sub-bandcontains phase noise itself).

The residual Tx-EEPN phase noise of Eq. (19) in the worst case (mostextreme sub-band) should be qualitatively compared with the overallLO-EEPN phase noise in a “full-band” coherent Rx (i.e., without using FBbased sub-banding).

The residual Tx-EEPN phase noise may be partially mitigated in our MSB(DFT-S) ODM by the carrier recovery system further downstream, justprior to the decision stage in each sub-band Rx. In our SC-MSBE Rx (FIG.40) the sub-bands are re-assembled into a complete single-carrierfull-band system and a block parallelized CR system (in the preferredembodiment of the MSDD type) is applied on the synthesized singlecarrier signal, thus the phase noise tolerance in this case should beidentical to that of a conventional SC Rx also using the same blockparallelized CR.

The method disclosed here of using a sparse pilot tone in themid-sub-band in order to demodulate the other sub-bands with it and thuscancel impairments such as the CFO and the non-linear phase noise, willalso work if the sparse pilot tone is transmitted in another sub-bandwith index i≠0 such that its center frequency does not coincide with thecenter frequency of the whole channel,

However, when using a sub-band other than the mid-sub-band for pilottransmission, the performance might be somewhat degraded as the maximalspectral distance between the pilot and each of the other sub-bands isincreased, and the sub-band farthest away may have less correlationbetween its phase and the phase of the sub-carrier.

Finally, the method disclosed here of using a sparse pilot tone in themid-sub-band in order to demodulate the other sub-bands at the output ofthe OS analysis FB should be compared with prior art usage of the sparsepilot tone to compensate laser phase noise and non-linear phase noise,as described in references [REF. A, REF. B] and other references therein. REF. A refers to Randel, S.; Adhikari, S.; Jansen, S. L.; “Analysisof RF-Pilot-Based Phase Noise Compensation for Coherent Optical OFDMSystems,” Photonics Technology Letters, IEEE, vol. 22, no. 17, pp.1288-1290, Sep. 1, 2010. REF. B refers to Liang B. Y. Du and Arthur J.Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,”,Optics Express, Vol. 19, pp. B862-B867 (2011).

In prior art, wherein FB based sub-banding is not used, the sparse pilottone (i.e., a pilot tone with null guardbands on either side) isextracted by means of a bandpass filter and used to demodulate theentire broadband optical channel. Thus a relatively narrowband pilotband is extracted out of the full bandwidth of the channel. The bandpassfilter (BPF) and the demodulator should in principle operate at asampling rate R equal to or higher than the bandwidth B of the entirechannel. In order to prevent loss of spectral efficiency, the pilotextracting BPF should be spectrally sharp, and relatively narrowband,but not too narrow, so as to capture the bandwidth of the disturbanceimparted upon the pilot tone in the course of optical propagation.

Initially, a pilot band of the order of tens of MHz was suggested in[Ref. A], aimed at mitigating laser phase noise, however in later works[Ref. B] the sparse pilot band was increased to the order of magnitudeof GHz in order to capture the non-linear phase noise spectral widtharound the pilot.

The problem with the prior work approach is the prohibitive complexityincurred in digitally implementing a sharp narrowband filter ofbandwidth of the order of B/M, where the bandwidth ratio M is typicallyin the range 10 . . . 20, operating at the full channel sampling rateR≧B. Because of the sharpness of the pilot extracting filter, a largenumber of taps, P, will be required in this filter, operating at fullrate R, thus the number of complex taps per second is PR.

It turns out that the complexity of this filter is much larger than thecomplexity of the polyphase filters array of a reference CS FB which hasits number of sub-bands set to be equal to M, and even larger than thecomplexity of an OS FB.

Indeed, comparing first with the CS FB complexity, a CS FB has Mpolyphase components each with P/M taps at the output of the S/P, i.e.,operating at the rate R/M. Thus, the total number of complexmultiplications per second in the polyphase array of the filter bankplus the IFFT is

${{M \cdot \frac{P}{M} \cdot \frac{R}{M}} + {( {\frac{M}{2}\log \; M} )\frac{R}{M}}} = {{{\frac{1}{M}{PR}} + {\frac{R}{2}\log \; M}} = {( {\frac{P}{M} + {\frac{1}{2}\log \; M}} )R}}$

This is the complexity of the CS FB to be compared with the complexityPR for the pilot-extracting BPF, i.e. the complexities ratio is

$\frac{{taps}_{CSFB}}{{taps}_{{PILOT}\mspace{14mu} {BPF}}} = {\frac{( {\frac{P}{M} + {\frac{1}{2}\log \; M}} )R}{PR} = {{\frac{1}{M} + {\frac{1}{2\; P}\log \; M}} \cong \frac{1}{M}}}$

and this ratio is much less than unity for typical values of M and Pwhich can be required to be of the order of tens of taps.

Notice that the complexity of an OS analysis FB will be even smallerthan that of the CS analysis FB, thus in turn the OS analysis FB issubstantially less complex than the pilot extracting FB. Moreover, inour solution for pilot extraction we just use one of the sub-bands ofthe OS analysis FB which is there anyway to perform the main function ofchannel processing for detection, therefore there is no incrementalextra complexity penalty in extracting the sparse pilot in our approach(as the FB hardware used to do it is already there). It follows thatusage of the OS filter bank entails a substantial complexity advantageover prior art extraction of the sparse pilot tone.

This completes the description of the mid-sub-band processor of FIG. 45providing inputs for CFO(+PN) estimation and compensation in all othersub-band processors. as well as CFO compensation via analog/digital PLLsto the LO laser/digital demodulator ahead of the OS analysis FB.

The invention may also be implemented in a computer program for runningon a computer system, at least including code portions for performingsteps of a method according to the invention when run on a programmableapparatus, such as a computer system or enabling a programmableapparatus to perform functions of a device or system according to theinvention.

A computer program is a list of instructions such as a particularapplication program and/or an operating system. The computer program mayfor instance include one or more of: a subroutine, a function, aprocedure, an object method, an object implementation, an executableapplication, an applet, a servlet, a source code, an object code, ashared library/dynamic load library and/or other sequence ofinstructions designed for execution on a computer system.

The computer program may be stored internally on a non-transitorycomputer readable medium. All or some of the computer program may beprovided on computer readable media permanently, removably or remotelycoupled to an information processing system. The computer readable mediamay include, for example and without limitation, any number of thefollowing: magnetic storage media including disk and tape storage media;optical storage media such as compact disk media (e.g., CD-ROM, CD-R,etc.) and digital video disk storage media; nonvolatile memory storagemedia including semiconductor-based memory units such as FLASH memory,EEPROM, EPROM, ROM; ferromagnetic digital memories; MRAM; volatilestorage media including registers, buffers or caches, main memory, RAM,etc.

A computer process typically includes an executing (running) program orportion of a program, current program values and state information, andthe resources used by the operating system to manage the execution ofthe process. An operating system (OS) is the software that manages thesharing of the resources of a computer and provides programmers with aninterface used to access those resources. An operating system processessystem data and user input, and responds by allocating and managingtasks and internal system resources as a service to users and programsof the system.

The computer system may for instance include at least one processingunit, associated memory and a number of input/output (I/O) devices. Whenexecuting the computer program, the computer system processesinformation according to the computer program and produces resultantoutput information via I/O devices.

In the foregoing specification, the invention has been described withreference to specific examples of embodiments of the invention. It will,however, be evident that various modifications and changes may be madetherein without departing from the broader spirit and scope of theinvention as set forth in the appended claims.

Moreover, the terms “front,” “back,” “top,” “bottom,” “over,” “under”and the like in the description and in the claims, if any, are used fordescriptive purposes and not necessarily for describing permanentrelative positions. It is understood that the terms so used areinterchangeable under appropriate circumstances such that theembodiments of the invention described herein are, for example, capableof operation in other orientations than those illustrated or otherwisedescribed herein.

The connections as discussed herein may be any type of connectionsuitable to transfer signals from or to the respective nodes, units ordevices, for example via intermediate devices. Accordingly, unlessimplied or stated otherwise, the connections may for example be directconnections or indirect connections. The connections may be illustratedor described in reference to being a single connection, a plurality ofconnections, unidirectional connections, or bidirectional connections.However, different embodiments may vary the implementation of theconnections. For example, separate unidirectional connections may beused rather than bidirectional connections and vice versa. Also,plurality of connections may be replaced with a single connection thattransfers multiple signals serially or in a time multiplexed mannerLikewise, single connections carrying multiple signals may be separatedout into various different connections carrying subsets of thesesignals. Therefore, many options exist for transferring signals.

Although specific conductivity types or polarity of potentials have beendescribed in the examples, it will be appreciated that conductivitytypes and polarities of potentials may be reversed.

Each signal described herein may be designed as positive or negativelogic. In the case of a negative logic signal, the signal is active lowwhere the logically true state corresponds to a logic level zero. In thecase of a positive logic signal, the signal is active high where thelogically true state corresponds to a logic level one. Note that any ofthe signals described herein can be designed as either negative orpositive logic signals. Therefore, in alternate embodiments, thosesignals described as positive logic signals may be implemented asnegative logic signals, and those signals described as negative logicsignals may be implemented as positive logic signals.

Furthermore, the terms “assert” or “set” and “negate” (or “deassert” or“clear”) are used herein when referring to the rendering of a signal,status bit, or similar apparatus into its logically true or logicallyfalse state, respectively. If the logically true state is a logic levelone, the logically false state is a logic level zero. And if thelogically true state is a logic level zero, the logically false state isa logic level one.

Those skilled in the art will recognize that the boundaries betweenlogic blocks are merely illustrative and that alternative embodimentsmay merge logic blocks or circuit elements or impose an alternatedecomposition of functionality upon various logic blocks or circuitelements. Thus, it is to be understood that the architectures depictedherein are merely exemplary, and that in fact many other architecturescan be implemented which achieve the same functionality.

Any arrangement of components to achieve the same functionality iseffectively “associated” such that the desired functionality isachieved. Hence, any two components herein combined to achieve aparticular functionality can be seen as “associated with” each othersuch that the desired functionality is achieved, irrespective ofarchitectures or intermedial components Likewise, any two components soassociated can also be viewed as being “operably connected,” or“operably coupled,” to each other to achieve the desired functionality.

Furthermore, those skilled in the art will recognize that boundariesbetween the above described operations merely illustrative. The multipleoperations may be combined into a single operation, a single operationmay be distributed in additional operations and operations may beexecuted at least partially overlapping in time. Moreover, alternativeembodiments may include multiple instances of a particular operation,and the order of operations may be altered in various other embodiments.

Also for example, in one embodiment, the illustrated examples may beimplemented as circuitry located on a single integrated circuit orwithin a same device. Alternatively, the examples may be implemented asany number of separate integrated circuits or separate devicesinterconnected with each other in a suitable manner

Also for example, the examples, or portions thereof, may implemented assoft or code representations of physical circuitry or of logicalrepresentations convertible into physical circuitry, such as in ahardware description language of any appropriate type.

Also, the invention is not limited to physical devices or unitsimplemented in non-programmable hardware but can also be applied inprogrammable devices or units able to perform the desired devicefunctions by operating in accordance with suitable program code, such asmainframes, minicomputers, servers, workstations, personal computers,notepads, personal digital assistants, electronic games, automotive andother embedded systems, cell phones and various other wireless devices,commonly denoted in this application as ‘computer systems’.

However, other modifications, variations and alternatives are alsopossible. The specifications and drawings are, accordingly, to beregarded in an illustrative rather than in a restrictive sense.

In the claims, any reference signs placed between parentheses shall notbe construed as limiting the claim. The word ‘comprising’ does notexclude the presence of other elements or steps then those listed in aclaim. Furthermore, the terms “a” or “an,” as used herein, are definedas one or more than one. Also, the use of introductory phrases such as“at least one” and “one or more” in the claims should not be construedto imply that the introduction of another claim element by theindefinite articles “a” or “an” limits any particular claim containingsuch introduced claim element to inventions containing only one suchelement, even when the same claim includes the introductory phrases “oneor more” or “at least one” and indefinite articles such as “a” or “an.”The same holds true for the use of definite articles. Unless statedotherwise, terms such as “first” and “second” are used to arbitrarilydistinguish between the elements such terms describe. Thus, these termsare not necessarily intended to indicate temporal or otherprioritization of such elements. The mere fact that certain measures arerecited in mutually different claims does not indicate that acombination of these measures cannot be used to advantage.

While certain features of the invention have been illustrated anddescribed herein, many modifications, substitutions, changes, andequivalents will now occur to those of ordinary skill in the art. It is,therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the invention.

We claim:
 1. A receiver that comprises: a first serial to parallelconverter arranged to receive first digital signals that representoptical signals of a first polarity, and to output the first digitalsignals via multiple outputs; a first polyphase finite impulse response(FIR) filter array coupled between the first serial to parallelconverter and to a first inverse fast Fourier transform (IFFT) module; asecond serial to parallel converter arranged to receive second digitalsignals that represent optical signals of a second polarity, and tooutput the second digital signals via multiple outputs; a secondpolyphase finite impulse response (FIR) filter array coupled between thesecond serial to parallel converter and to a second (IFFT) module; andmultiple sub-band processor modules; wherein each sub-band processormodule is coupled to inputs of a same order of the first and second IFFTmodules.
 2. The receiver according to claim 1 wherein the sub-bandprocessor modules are Orthogonal Frequency Division Multiplex (OFDM)sub-band receiver modules
 3. The receiver according to claim 2, furthercomprising a de-mapper and data multiplexor that follows the multipleOFDM sub-band receiving modules.
 4. The receiver according to claim 1,further comprising a coherent optical front end and two pairs of analogto digital converters (ADCs); wherein the coherent optical front end isarranged to receive the optical signals of the first and secondpolarity; provide analog signals representative of the optical signalsof the first polarization to a first pair of ADCs, and provide analogsignals representative of the optical signals of the second polarizationto a second pair of ADCs; wherein the first pair of ADCs is coupled tothe first serial to parallel converter; and wherein the second pair ofADCs is coupled to the second serial to parallel converter.
 5. Thereceiver according to claim 2, wherein each OFDM sub-band receivingmodule comprises: two sequentially coupled sets of components, eachsequentially coupled set of components comprises a sub-band impairmentcompensator, a serial to parallel conversion and cyclic prefix dropmodule, a 2N-points fast Fourier transform (FFT) module and a half banddecimator; a pair of parallel to parallel converters; a plurality ofdual input dual output (2×2 MIMO) equalization modules, whereindifferent pairs of outputs of the half band decimators for the twopolarizations are coupled to pairs of inputs of different 2×2 MIMOequalization modules; wherein each 2×2 MIMO equalization modulecomprises a pair of outputs that are coupled to inputs of a same orderof the pair of parallel to parallel converters; and a pair of carrierrecovery and decision modules, each carrier recovery and decision moduleis coupled to a parallel to parallel converter of the pair of parallelto parallel converters.
 6. The receiver according to claim 2 wherein thesub-band processors comprise two sequentially coupled sets ofcomponents, each sequentially coupled set of components comprises asub-band impairment compensator, a serial to parallel conversion andcyclic prefix drop module, a 2N-points Fast Fourier transform (FFT)module and a half band decimator; a pair of parallel to parallelconverters; a plurality of dual input dual output (2×2 MIMO)equalization modules, wherein different pairs of outputs of a same orderof the half band decimators for the two polarizations are coupled topairs of inputs of different 2×2 MIMO equalization modules; wherein each2×2 MIMO equalization module comprises a pair of outputs that arecoupled to inputs of a same order of the pair of parallel to parallelconverters;
 7. The receiver according to claim 6 wherein the carrierrecovery and decision modules are multiple-symbol differential detection(MSDD) decoders,
 8. The receiver according to claim 6 wherein eachsub-band impairment compensator comprises the cascade of an in-phasequadrature imbalance (IQI) compensator, a mixer and a delay unit.
 9. Thereceiver according to claim 9 wherein each sub-band impairmentcompensator comprises a cascade of an in-phase quadrature imbalance(IQI) compensator, a mixer and a delay unit.
 10. The receiver accordingto claim 6 wherein the half-band decimator is filterless, and isarranged to route an N-points input either into a high or a lowhalf-band of N points out of 2N inputs, dropping the remaining N points.11. The receiver according to claim 9 wherein the half-band decimator isfilterless, and is arranged to route a N points input either into a highor a low half-band of N points out of its 2N inputs, dropping theremaining N points.
 12. The receiver according to claim 6 wherein thehalf-band decimator is arranged to perform a cyclic shift for oddnumbered sub-band indexes module.
 13. The receiver according to claim 9wherein the half-band decimator is arranged to perform a cyclic shiftfor odd numbered sub-band indexes module.
 14. The receiver according toclaim 6 wherein the sub-band processors are arranged to feed an array of2Nssc IFFT modules wherein Nssc IFFT modules form a sub-arraycorresponding to X polarization of signals and Nssc IFFT modules form asub-array correspond to Y polarization of signals; the number Nsscdivides the number P of sub-band processors; The sub-band processors arepartitioned into P/Nssc groups; the X-outputs of the sub-band processorsin the g-th group are uniformly spread onto the inputs of the g-th IFFTin the X sub-array via serial to parallel converters; the Y-outputs ofthe sub-band processors in the g-th group are uniformly spread onto theg-th IFFT in the Y sub-array via serial to parallel converters. Thenumber of output ports of the serial to parallel converters used toperform the spreadings is given by the IFFT size divided by the numberof sub-band processors.
 15. The receiver according to claim 14 whereineach sub-band processor module comprises: two sequentially coupled setsof components, each sequentially coupled set of components comprises asub-band impairment compensator, a serial to parallel conversion andcyclic prefix drop module, a 2N-points fast Fourier transform (FFT)module and a half band decimator; a pair of parallel to parallelconverters; a plurality of dual input dual output (2×2 MIMO)equalization modules, wherein different pairs of outputs of the halfband decimators for the two polarizations are coupled to pairs of inputsof different 2×2 MIMO equalization modules; wherein each 2×2 MIMOequalization module comprises a pair of outputs that are coupled toinputs of a same order of the pair of parallel to parallel converters;16. The receiver according to claim 6, further comprising a joint IQI,Carrier Frequency Offset (CFO), Coarse Timing Offset (CTO), SamplingFrequency Offset (SFO) estimation module that is coupled to the mixerand to the delay unit of each sequentially coupled set of components.17. The receiver according to claim 6, further comprising a joint IQI,Carrier Frequency Offset (CFO), Coarse Timing Offset (CTO), SamplingFrequency Offset (SFO) estimation module that is coupled to the mixerand to the dealy delay unit of each sequentially coupled set ofcomponents.
 18. The receiver according to claim 6, further comprising apair of IFFT modules that are coupled between a dual input dual output(2×2 MIMO) sub-band receiving modules and the pair of parallel toparallel receivers.
 19. The receiver according to claim 18 wherein eachof the two FFT modules has 2N points and each of the two IFFT moduleshas N points.
 20. The receiver according to claim 6, further comprisinga pair of IFFT modules that are coupled between the MIMO sub-bandreceiving modules and the pair of parallel to parallel receivers. 21.The receiver according to claim 1, further comprising a middle sub-bandprocessor that is arranged to process information that belongs to acentral sub-band.
 22. The receiver according to claim 21, wherein themiddle sub-band processor is arranged to provide Carrier FrequencyOffset (CFO) signals and Phase Noise (PN) estimation signals to the OFDMsub-band receiving modules.
 23. The receiver according to claim 1,wherein there are multiple (M) sub-bands and wherein there are M−2 OFDMsub-band receiving modules.
 24. The receiver according to claim 1,wherein each OFDM sub-band receiving module comprises two sequentiallycoupled sets of components, each sequentially coupled set of componentscomprises: an impairments recovery module; a serial to parallelconversion and cyclic prefix drop module; an 2N point fast Fouriertransform (FFT) module; a half band decimator; an N-point IFFT module; aparallel to serial converter; a multiple input multiple output (MIMO)equalization module; a serial to parallel converter; an N point FFTmodule; a cyclic shift for odd numbered sub-band indexes module; a 2Npoint IFFT module; and a parallel to serial converter.
 25. The receiveraccording to claim 24, wherein the impairments recovery module comprisesan in-phase quadrate imbalance (IQI) compensator, a mixer and a delayunit.
 26. The receiver according to claim 1, wherein each OFDM sub-bandreceiving module comprises two sequentially coupled sets of components,each sequentially coupled set of components comprises: an in-phasequadrate imbalance (IQI) compensator; a mixer; a delay unit; a serial toparallel conversion and cyclic prefix drop module; an 2N point fastFourier transform (FFT) module; a half band decimator; a parallel toserial converter; a multiple input multiple output (MIMO) equalizationmodule; a serial to parallel converter; a cyclic shift for odd numberedsub-band indexes module; a 2N point IFFT module; and a parallel toserial converter.
 27. The receiver according to claim 1, furthercomprising a pair of IFFT modules, a pair of parallel to parallelconverters and a pair of block-parallelized carrier recovery (CR) anddecision modules.
 28. The receiver according to claim 27, wherein eachOFDM sub-band receiving module comprises: two sequentially coupled setsof components, each sequentially coupled set of components comprises anin-phase quadrate imbalance (IQI) compensator, a mixer, a delay unit, aserial to parallel conversion and cyclic prefix drop module, a fastFourier transform (FFT) module and a half band decimator; a pair ofparallel to parallel converters; and a plurality of multiple inputmultiple output (MIMO) equalization modules; wherein different output ofdifferent half band decimators are coupled to inputs of a same MIMOequalization module; wherein each MIMO equalization module comprises apair of outputs that are coupled to inputs of a same order of the pairof parallel to parallel converters.
 29. The receiver according to claim1, further comprising at least four sequences of modules, wherein eachsequence of modules comprises: an IFFT module; a parallel to parallelconverter; and a block-parallelized carrier recovery (CR) and decisionmodule; wherein different subsets of inputs of each IFFT module arecoupled to different OFDM sub-band receiving modules.
 30. The receiveraccording to claim 14 wherein a central frequency sub-band of each ofthe two oversampled analysis filter banks is passed to a mid-sub-bandprocessor that is arranged to extract estimates of common phaseimpairment of all the sub-bands to be fed forward to all the sub-bandsfor common phase estimation and also is arranged to extract feedbackestimates of carrier frequency offset (CFO) to be fed to frequencycorrection means in the receiver.
 31. The receiver according to claim30, arranged to receive feedback from a mid-sub-band processor that is adigital demodulation module located at the analysis filter bank inputs,consisting of complex multiplication of each of the polyphase filterarray inputs by the CFO control signal passed by the mid-sub-bandprocessor.
 32. The receiver according to claim 30 wherein each centralsub-band passed to a mid-sub-band processor for each polarization ispassed through an IQI compensation module with IQI parameter estimatedas the mean value of two neighboring sub-bands on either side of thecentral sub-band.
 33. The receiver according to claim 30 wherein the IQIcompensated received central sub-band for either a X or Y polarizationis passed through an angle extract module to extract a common phaseestimate to be fed forward to CFO and NL phase noise compensators ofother sub-bands.
 34. The receiver according to claim 33, wherein theextracted angle is passed to one or two phase locked loops feedingeither the Local Oscillator laser frequency control and/or ademodulation module ahead of the analysis filter bank for thatpolarization.
 35. A transmitter that comprises: a set of first filterswith outputs connected to a summing node; a set of upsamplers; a set ofsub-band processors, wherein each sub-band processor comprises aninterpolator, a cyclic shifter and a second filter; wherein each cyclicshifter is coupled between an interpolator and an upsampler; whereineach first filter follows an upsampler; wherein different interpolatorsare arranged to receive different sets of signals and performup-sampling; wherein the set of first filters is arranged to outputvirtual sub-channels of information occupying disjoint spectralsub-bands; wherein each sub-band is associated with a pair of filtersthat comprises a first filter and a second filter, wherein the firstfilter has a milder frequency response outside the sub-band than afrequency response of the second filter outside the sub-band.
 36. Thetransmitter according to claim 35, wherein each second filtersubstantially nullifies spectral components outside the sub-bandassociated with the second filter; wherein each first filter passesspectral components that belong to at least one sub-band that differsfrom a sub-band associated with the first filter.
 37. The transmitteraccording to claim 35, wherein each upsampler performs a L-factorupsampling, and wherein each interpolator performs a V-factorupsampling; wherein L and V are positive numbers.
 38. The transmitteraccording to claim 35, wherein L and V differ from a number (M) of thesub-bands.
 39. The transmitter according to claim 35, wherein V equals4/3.
 40. The transmitter according to claim 35, wherein eachinterpolator comprises: a serial to parallel converter; a N point fastFourier transform (FFT) module arranged to output N element vectors; azero padding and circular shift module arranged to perform a zeropadding operation and a circular shift operation on the N elementvectors to provide V*N element circled vectors; a N*V point inverse FFT(IFFT) module; and a parallel to serial conversion and cyclic prefixadder coupled to an output of the N*V point IFFT module.
 41. Thetransmitter according to claim 40, wherein the zero padding and circularshift module is a routing fabric arranged to implement the circularshift operation and the zero padding operation by performing a mappingbetween outputs of the N point FFT module and inputs of the N*V pointIFFT module.
 42. The transmitter according to claim 41, wherein therouting fabric is arranged to implement the circular shift operation andthe zero padding operation without storing any elements of a N elementoutput vectors within a buffer and without performing data transfersbetween different locations of the buffer.
 43. The transmitter accordingto claim 42, wherein the routing fabric is arranged to couple betweensome groups of outputs of the N point FFT module and some groups ofinputs of the N*V point IFFT module.
 44. The transmitter according toclaim 35, wherein each virtual sub-channel of information that occupiesa sub-band is an Orthogonal Frequency Division Modulation (OFDM)compliant sub-channel of information; wherein each interpolator iscoupled to an OFDM transmitter module; wherein a combination of eachinterpolator and OFDM transmitter module comprises: a serial to parallelconverter; a zero padding and circular shift module arranged to performa zero padding operation and a circular shift operation on outputvectors of the serial to parallel converter to provide sero padded androtated vectors; an N*V point inverse FFT (IFFT) module; a parallel toserial conversion and cyclic prefix adder coupled to an output of theN*V point IFFT module.